There are a wide variety of champions in LoL right now. I'm finding they fit a broader mold than the standard "AD Carry / AP Carry / Bruiser / Support" buckets that people tend to put them in. Not a lot of people are writing about this and i wanted to give it a try. Let's assume you're "newish" to League of Legends and you've bought the two champions bundles. That gives you a stable of 40 champions to choose from, each with a pretty wide variety of strengths and weaknesses. I wanted to give a quick breakdown on WHEN you should choose each of those champions and what you should look out for as well as possible counterpicks to avoid. this will be a fairly long post so I'd advise reading it as more of a reference:
Alistar: Alistar typically gets played as an aggressive support, a jungler or, occasionally in mid lane solo. He's very tanky and designed to get in people's faces. He's ideally suited for playing with ranged AD Carrys where he can run interference in front of them and provide knockbacks and knockups (that your AD carry can continue to shoot at). If you use him, try to initiate with headbutts from the side so that your AD carry can continue to shoot at the target and then move for the pulverize for additional CC. Your spam is spammable but don't waste too much mana on it early on.
Amumu: A very good jungler in season 2. He's especially a good choice if facing high hp opponents. His ganks are strong but his bandage toss skill shot is slow and somewhat tricky to hit with. All of his damage is AoE around him so you'll want to be built a little tanky to take the inevitable damage you take. Also, make sure you're familiar with the jungle and what's expected before you start running him.
Anivia: In pro play, he's exclusively run as an AP mid but i think there's no reason you can't run him as an aggressive support with slows and the wall. If you do, he's probably best with a melee character who will get the biggest boost from you having those two abilities. Try to get to where you can quickly cast the wall to break two enemies apart. This can be really powerful if you trap a carry or melee character on the wrong side of a 5 second wall. Note, if you think you're going up against a champion with a repeatable flash affect (Ezreal, for example) then your wall will be significantly less useful with that character so try to catch out other characters or pick someone other than Anivia.
Annie: Pretty much the definition of AP. She's very bursty. With proper timing she can get a large cone aoe stun with her incinerate synergizing with her passive. I don't see her get played a lot and I'm honestly not sure why. She's tankier than most AP Carries, has spammable abilities and a really nice team utility (aoe stuns??). Her mana costs are a bit on the high side but other than that, it's hard to find fault with her. One note about her Q. The mana cost is refunded if she kills her target and it only has a 4 second CD. This makes it perfect for farming and last hitting with.
Ashe: Typically built as an AD carry. The thing about her is her ability to slow you with regular shots. Keep this in mind and recognize that champions that rely on mobility are going to be at a slight disadvantage. Her cone is a big aoe spread but hits the first target so you can make it useless with minions (same with her ultimate... or you could just sidestep it). She has a vision utility power that you can't do anything about but you can keep in mind as well.
Blitzcrank: Another aggressive support. Blitzcrank's signature ability is his skill shot which will grab you, pull you close to him and then stun you. When picking him, you probably don't want to be up against the more aggressive melee supports (Leona, Taric, Alistar). He works well with a melee or a ranged AD though you'll need better team communication if a melee carry is in his team. He can keep a character locked down for 2+ seconds with his grab and then followup Power Fist. He's designed to support a kill lane, however. If your carry is a late game character (Graves, Ezreal, etc), he's not nearly as ideal.
Cho'Gath: Lately, he's seen pretty common use as a jungler. His kit is unique. He has a very powerful, but somewhat hard to hit, delayed cast aoe knockup as well as a aoe cone silence. What really sets him apart is the fact that he'll grow to be gigantic through his Feast ultimate. He can add 900 health this way with a full six stacks. You want to make sure that this is nearly always maxed and you also want to make sure it's up for Team Fights as it will deal a HUGE chunk of true damage to a champion (650 + .7x AP). He has very good sustain with health and mana getting returned with each kill (minion, monster or champion). Ganking with him is as simple as hiding in a bush, waiting to land rupture and coordinating with your laner.
Corki: A very popular AD carry. The real power he has is in his spammable AoE ultimate. He's far and away the most "aoe" focused AD carry with Gatling Gun and his ultimate allowing for AD scaling AoE damage and also phosphorous bomb scaling off AP (an aoe Damage and reveal). He's slow and has no CC so a support that covers that deficiency and/or an opponent without a lot of mobility is a positive.
Dr. Mundo: Gets used as a jungler alot but I'm not a big fan. He's tough to kill and has a deceptively slow cooldown machete throw that serves as a slow. It's a sufficient initiate but not nearly as good as Blitzcrank or Cho'Gath. He has a PBaOE untargeted ability that does well later... especially if you also build a Sunfire cape. Did not perform well in the latest tournament. It remains to be seen if he'll fit in this newest jungle or not.
Evelynn: An assassin par excellence. She is typically seen as an AP mid. If you want to watch how she should be played, watch Alex Itch playing her for Moscow Five. He's scary. She's a somewhat fragile champion and you need to learn exactly how much damage you can do in a burst with her because if you can't kill her opponent quickly, you're probably going to die yourself. One of only a few characters with built in stealth capability, she's very useful if up against relatively squishy targets that are likely to push. Catching them out of position becomes almost easy. Watch out for pink wards being purchased.
Fiddlesticks: Originally a very good jungler. He fell off in Season 2 due to the perception that he had insufficient AoE capacity (which is debatable with Dark Winds). I've run him through the new jungle and, even at my summoner level of 15, he appears to be able to jungle just fine. I cleared the jungle with no support in 4 minutes. With a bit of leashing help on golem/wolves, that would easily have been in the 3:30 range which is more typical. His ganks are ridiculous when he hits 6. You should crowstorm in slightly behind your opponent, terrify, dark wind and then drain. This will be insufficient to kill most opponents but with ANY kind of support, it will get you one to two kills. Some people play him as a pure support for AD Carry's. I don't know much about this but the multi-target silence and terrify give you sufficient CC that i can see this being a reasonable alternative.
Gangplank: Yarrggh! He's a pirate. You play him when you need a pirate. He accumulates gold quite quickly with "Parrley". Some people have already begun to talk about a Bankplank build that uses Parrley and Pick Pocket to accumulate massive amounts of gold while playing a sort of harassing support. I think his general kit is a little niche at present (self heal, aoe attack speed + movement speed buff are somewhat niche powers). His ultimate, however, covers a very large area and is great when either sieging the enemy base or fighting off a siege of yours.
Heimerdinger: He's a funny little guy. I actually think one of the new changes in the meta may give him a role. I want to play some games and confirm this, but I suspect he'll be very good in a 1v2 role if left up top (or on the bottom against a support/AD Carry combo). Be sure if fighting against AoE champions that you put your turrets out wide to minimize the AoE damage but put them clustered in the minion line if they have single target skill shots (like Ezreal's Q). Also, their range is quite good at 525. This makes him a good choice against Graves (525), Kog-Maw (500), Sivir (500) and Urgot (425). He's not bad against Ezreal, Miss Fortune, Corki or Tristana who all have 550 range either. Two other notes, his W auto-targets the three closest enemies with a HUGE (1000) range. This lets you stand very far back and hit people with impunity. It will NOT let you fire if there's no one in range so you can use that to judge distances. Also, his grenade is hard to hit with but when it does, it gives fairly decent damage and a decent length hard stun. Keep this for when your jungler is ready to gank and try to help them out when the time is right. Note: He can support given his stun and aoe heal. If you do this, be stingy with your turrets as they suck a share of xp and can result in a starved carry if you're not careful.
Janna: Another support. While she has some aggressive capacity, she also has more "standard" support abilities with a shield and an ultimate that is almost entirely support in nature. The trick to her is maximizing her knock up. It's very difficult to use against fleeing opponents and much better to use as an initiate. This means you want throw it out there at roughly the same time as your AD Carry initiates. Her shield works perfectly with melee characters by both shielding as well as buffing AD. This allows her to throw the hurricane, shield and then slow an opponent all in a relatively short time to, hopefully, setup a kill.
Jax: One of two melee "noobstompers" in the game (along with Master Yi). He is a beefy snowballing machine. I'd tend to use him if you know you're up against a similarly beefy solo top laner or a squishier assassin type that you can out sustain with your attack speed scaling. In a pinch, he can also do a decent initiate with his leap strike and a prematurely activated counter strike. That can be used to start a gank if your jungler lacks a good initiate.
Kayle: I like her but I'm not totally comfortable playing her. Her heal lacks the oomph to really give her sustain in lane although it's secondary ability is quite good (allowing her to get BACK to lane quickly or give someone that speed boost needed to finish a fleeing opponent). her Q and E abilities are great, however. Late game she can clear waves very, very quickly and her ultimate allows her to keep anyone safe in a team fight for 3 seconds. Her ability to debuff champions combined with her E ability makes her very good at softening team targets late game, as well. In the current meta, she works well as a support or as a solo top laner and (I think) she can handle the 1v2 situations that the latter is likely to entail frequently.
Karthus: Almost always played as an AP mid. The key to him is to watch your opponents and look for making his ult matter. The other thing is to not be afraid to die. You have a PBAoE big time damage spell that continues on when you die. Make yourself just a little bit beefy and don't be afraid to die in a team fight. Follow that up with a Requiem (don't get interrupted!) and you can manage to get an awful lot of kills and assists.
Kassadin: This guy is a hard core mage assassin but he is all about the late game. Because his ultimate starts at a relatively long CD, he can't really do his thing (chain Rift Walks) until he's gotten it to at least lvl 2 and realistically lvl 3. If you're going to go this route, you need to have crazy high mana regen so that you can get the rift walk up to 4-5 stacks (at which point it will cost you 400-500 mana but also be doing nearly 800 damage). The tooltip says stacks 10 times. I don't think anyone realistically gets to that, however. Shoot for 4-5 chaining up to 6-7 as you stalk an opponent. His Q also gives him a relatively long duration, hard hitting silence as well. You probably also want to talk to your jungler and make sure it's okay for you to take blue buff as you're going to need the mana. At least you can get there and back quickly with Rift Walk.
Katarina: The poor man's Diana, at least that's how I've started thinking of her. She is significantly easier to get around with because her Shunpo is more flexible (anyone... not just enemies) and a lower CD (without Crescent Strike). She's also a better AoE damage dealer. But she trades out having the durability of Diana as well as the utility and sheer raw damage. Still, I've seen people snowball hard with her. She's especially deadly against more squishy mages in the middle (Twisted Fate, Karthus, etc).
Malphite: He's big and beefy. You're going to build armor and hit points almost exclusively with him. His initiate is, by far, the best in the game with a large AoE charge that results in a big knock up. With a modest amount of coordination, you can have your team killing people before the other team hits the ground. Don't focus on getting the whole team in the AoE. Focus on getting the right one or two people isolated. If you can drop 2 people in the first few seconds of a team fight, you're going to win. Your job with Malphite is to start those fights and take damage. I typically build Frozen Heart, Sunfire Cape, Abyssal Scepter, Aegis of the Legion and Guardian Angel. That insures that I'm very, very difficult to kill while also allowing me sufficiently decent damage output to where I can't be completely ignored like some tanks can. Remember that your Seismic Shock siphons speed (slowing an opponent and speeding yourself). If you hit someone with a phantom dancer, you WILL go very fast. Your ground slam decreases attack speed and late game will do ~400 damage so use it immediately after charging in.
Master Yi: The other "Noobstomper". Master Yi is very, very dangerous once he gets a little bit fed. He's been used to good affect against squishy AP Mids as well as top lanes that lack escapes. Look to trade kills if you have to. You're going to scale better with kills than most people will.
Morgana: She's a bit of an odd duck. Kind of an early game AP Mid that transitions into a late game support. Her abilities are on very long cooldowns and her built in spell vamp means that early game her sustain is decent but late game (when most everyone else's cooldowns are declining) she's relegated to using her shield more. She's also best when supporting an AP carry and not an AD carry because her aoe lowers magic resist. If you're not running a jungler, she's a good top support for an AP bruiser / AP Carry.
Nasus: This guy is the definition of the solo top bruiser. You are going to want to farm, farm, farm with him and get that Q up. If you're going to be harassed, you don't want him. I would never pick him in draft but against another top lane champion with relatively limited ranged attacks (Malphite, Rengar, Olaf, Shen, Jayce even before 6). You have no poke but your spirit fire is not insignificant and your wither will make ganks easier. Just make sure you keep last hitting that Q. You want your damage with it to be 600+ by end of the game.
Nunu: He has gotten played solo top, jungle and now, frequently, support. He's another aggressive support. The combination of blood boil and ice blast helps to make your AD carry very scary. On the other hand, consumer is a fantastic tool for maintaining your fighting form up top or in the jungle. If Jungle transitioning into Support is a thing your team is going for, he could be very good. Note, he's also a very good counter pick for anyone with pets (Yorick, Malzahar, etc) because his consume will take off significant portions of their DPS if times right.
Rammus: He's a little rolling ball of armored death. His Q will chase someone down with frightening inevitability and knock them up. It's a great lead in for your AD carry. You can then go defensive and taunt which has the added benefit of lowering the opponents armor as well. Again, another aggressive support and one that we haven't seen enough of, I think. Dropping an opponent's armor to essentially zero early game can be devastating. You should look to focus on W and E if you're playing support.
Ryze: Lately, he's started seeing play as a counter to Diana. I'm not entirely convinced he does that but the pros seem to have faith in it. Ryze scales off his mana pool making mana runes, masteries and items really important. The important thing to remember with him is that you want to go all out. A lot of other characters have to conserve their mana early. But Ryze gets better with more mana and better with more casts. So cast everything and cast it often.
Shaco: I don't know a lot about Shaco. He's a hard core assassin which hasn't been fitting in the meta game very well lately. Some people use him as a jungler but that doesn't seem to be very common. He's tricky in that he produces little jack in the boxes that fear while he also hops around and backstabs you. In reading about him, he somewhat lacks the burst potential of other assassin champions and so has been a bit eschewed by the pro community.
Singed: A very odd character. None of his abilities feel like normal abilities for LoL. He doesn't shoot anything at you. He is more of a strategic player with a potent acid trail and a powerful fling. His abilities all scale really well off AP but typically, i think he gets build AD because his only real attack in combat is his autoattack. His slow is the strongest slow in the game (75%). You want to make good use of his poison trail and slow as well as his fling. Look for opportunities to fight under your tower and fling your opponent into your tower. Not a champion I'd suggest for your first game, however.
Sion: The undead lumberjack. He's a melee beast with a power that drains his health and ups his damage but also permanently increases his health with each unit kill. His passive gives him strong resistances to AD carries (at least early) and he can be a pretty hard counter to them. If a team looks like they're picking an AD heavy champion for top (Yorick, Rengar, etc) he could work to apply very strong early pressure to them.
Sivir: Widely acknowledged as the best lane pusher in LoL. Her Q and W allow her to clear minion waves better than almost anyone. Her spell shield is tricky to master but can be used powefully against certain people. Anyone with an AoE over time ability (Miss Fortune, Ziggs, Gangplank, etc) can look forward to her nullifying a tick of it and gaining 75 mana. It's powerful against anyone with a predictable ability (Cho'Gath's feast getting shielded away is fun). Because it stops an ability cold (without regards to damage) it's nice against any channeled ultimate you can see coming (Karthus, Nunu, etc). Her ultimate will greatly benefit other attack focused champions so teaming her with a top lane Lee Sin and a AP Mid Teemo can be brutal.
Soraka: The most passive of supports (outside of Lulu perhaps). She really is what you traditionally think of as support. She can heal, give mana, buff armor, debuff MR and her passive buffs nearby MR. Her ultimate lets her heal everyone on the map (her teammates at least). She's support all the way. Late game, her starfall will greatly help the AP Carry to deal damage (and possibly the bruiser top if he's someone like malphite who wants to deal magic damage). There's a reason that she currently has the highest win rate of any champion on the NA servers. If you play her, it may feel like you're taking a bullet... but you're REALLY helping your team.
Taric: Yes another aggressive support. With a stun and the ability to replace his own mana and heal two people (himself and a friend), he can help a carry stay in lane a long time. The fact that he has high armor and he boosts nearby armor of allies passively just puts icing on the cake of him being a great AD Carry support. It's not hard for a Taric / Graves combo to have the AD carry with 60+ armor during early fights. That's a 38% reduction in damage taken which is huge in early fights.
Teemo: There are two ways to build teemo. AP + Attack speed or AD. Of those two, most people greatly prefer the first. Some people feel its cheesy. Other people like putting invisible poisonous mushrooms all over the map. The people that play Teemo are the latter. His unique, when pumped with sufficient AP is scary. Those mushrooms also serve as mini-wards which is not an insignificant bonus.
Tristana: I know very little about her. She used to be a very popular AD carry. Her auto-attack increases in range until its just over 700 at lvl 18. That's a unique ability that no other champion has (Xerath has an activated range boost for his spells). It also makes her the longest range AD carry from levels 12 on. If I'm understanding her passive correctly, she also turns a wave of minions into a liability for the enemy (exploding minions cause damage to other minions that makes them explode ad infinitum. You could build her as a straight AD carry or you could mix in some AP items (Hextech Gunblade and/or Guinsoo's Rageblade). Also interesting would be Runaan's which would put you pretty close to the attack speed cap when you activate Rapid Fire (2.5 attacks per second). Given the long range, that could be a real advantage in team fights.
Tryndamere: I hate this guy. He's a ridiculously invincible team fighting machine. He has an unCCable ult that makes him invulnerable to death for 5 seconds (on a relatively short, 90 second cooldown). You want to stack ridiculous amounts of crit, AD and little bits of attack speed on him. He'll be more powerful earlier in smaller team fights where you don't have to worry about getting burst down despite without a chance to pop your ult. He's also better against AD heavy teams with a more dependable (i.e. timeable) damage stream.
Twisted Fate: Probably one of the highest (if not the highest) skill cap character in the bundles. Twisted Fate has really amazing lane presence through his ability to clear minions easily and sustain through blue cards, harass through gold cards and aoe damage with reds. There are a number of ways to build him as his abilities scale well enough off AD that you can go that route. Most people choose to go AP, however. One note about his ultimate. It takes two button pushes to get it off. The first grants stealth detection and the second lets him teleport in a pretty wide (but not endless) area. MOST people send him mid now so that he can take a few steps out of lane and get to the top or bottom lanes with the teleport. Don't discount the stealth detection as it can help against people like Teemo, Evelynn and others with built in stealth abilities.
Twitch: I like Twitch a lot. He applies a True Damage DoT that can be pretty brutal and his abilities synergize very well. I think the new Runaan's Hurricane is an item made almost especially for him and gives him a powerful presence in team fights (allowing expunge to be far, far more effective). The True damage is especially nice if you're up against opponents that are likely to stack MR (think Galio) and Armor (Malphite, Xerath).
Veigar: The AP Carry version of Nasus. You want to farm like a champ and get your Q powered up to an insane level during the laning phase. He has very good synergy with certain supports (Morgana!) and Junglers (Amumu!) but can be a little fragile. I'd use him if you have opponents that are a little squishy and/or lack flash like escapes (Graves, Corki, etc). His AoE stun can be very powerful against those types of characters. In conjunction with a character like Trundle, Ziggs, Anivia, Caitlyn or others that have the ability to mine out certain areas, you can really control the shape of the battlefield where you have your team fights.
Warwick: Great sustain and the ability to either jungle or top lane makes this one of the more flexible champions. He hasn't seen a lot of play at the pro-level because his jungle time hasn't been very good but with the Jungle changes, I suspect that will change. His jungle ganks can be terrifying and, once he starts accumulating attack speed and AD, he can carve through enemy champions very quickly. Be careful about using him in the jungle if you have an AD Bruiser in the top lane as this could make your team somewhat unbalanced towards physical damage. A very good option if you're facing a slower than normal team too (particularly one without the ability to control the field of battle like Anivia).
Zilean: The oddest support in the game, IMO. His early game burst damage is ludicrous. It's not uncommon for a time bomb to hit someone for 25% of their health early on. Late game, he becomes more difficult. His "support" is in a difficult to time ultimate. He has a dual purpose slow/speed boost but it's on a lowish CD. His CDs in general are very long (although he has an ability that will shorten them). He actually has good speed (310) and good range (600). I'm curious if anyone has tried to grow him into a hybrid AD carry (with Hextech Gunblade and Guinsoo's Rage Blade). He would never have the raw damage of someone like Twitch but it seems like it might catch some people by surprise.
Anyways... I know that was long but it should give you some thoughts on how to build/pick/maintain a good team comp with these champions that almost everyone should have.