Kiwi - League of Legends

It's really surprising how many people is not aware of their role as they play, you'll see supports kill stealing, and last hitting, tanks who forget to protect the carry and just charge and try to get kills, and more, in this entry I'll just name the most common roles in LoL and what they are expected to do.

AP Carry
Abiliy power carry, your role is to deal lots of burst magic damage, then back, then burst again, you normally just sit behind your tanks so you can unload safely, sometimes you can initiate too, for example, dropping a good tibbers is always nice.
Common summoner spells: Flash & Ignite
Some good AP Carries: Annie, Brand, Vladimir, Orianna

AD Carry
Attack damage carry, normally this refers to ranged AD carries, your role is stay back and poke, deal damage constantly during the whole teamfight, normally bruisers will try to get you, but it's up to the tanks and supports to protect you, you can deal the most consistent damage if fed. It's important to not back too much, you need to do damage, many carries fail because they are too scared to go in, and their team end up getting killed.
Common summoner spells: Flash & Ghost
Some good AD Carries: Ashe, Caitlyn, Vayne, Tristana, Teemo

Junglers
Junglers are very important, they allow top lane to solo, while they clear the jungle and control buffs and dragon, jungling is a very deep matter itself, but for now I'll say the basics, junglers are supposed to gank each overextending lane, or help a teammate who is losing, you can gank for the kill, or for the summoner (most commonly flash), so you can gank again later, or any other gank, they should play safer when they burn they summoner, giving your partner an advantage.
Junglers are normally bruisers or tanks, which once they finished jungling they should play their respective role.
Common summoner spells: Flash/Ghost & Smite
Some good Junglers: Udyr, Amumu, Warwick, Xin Zhao, Nunu

Support
Support is a rather easy role, but surprisingly there's a lot of misunderstanding and fail on this role, support have to... support your carries, this is, heal them, CC if they try to gank them, die for them sometimes, help them last hit under the tower, CV everywhere on the map, find the enemy jungler, find the enemy team, etc.
Supports normally just sit bottom lane with a ranged AD carry.
Common summoner spells: Flash/Ghost & Clarivoyance
Some good supports: Sona, Soraka, Janna, Taric

Tanks
Tanks main role is to protect the carries, initiate and get focused. Some tanks go around in a teamfight trying to get kills, and completely forget about their carry, this is really bad, you want to protect them, and let them get the kills most of the time.
Common summoner spells: Flash & Ghost
Some good tanks: Singed, Alistar, Udyr, Amumu, Galio

Bruisers
Bruisers, also known as tanky DPS, are mostly meelee champions with a lot of sustain and life, bruisers normally focus the enemy carry, and take them out of the fight, or kill them, fed bruisers can carry pretty hard.
Common summoner spells: Flash, Ghost, Ignite
Some good bruisers: Jax, Warwick, Xin Zhao, Lee Sin, Irelia

Assasins
Assasins are champions with very good chasing skills, and a lot of burst, they can blow up the enemy carries, and normally can engage and disengage easily. Assasins do mostly magic damage.
Common summoner spells: Flash & Ignite
Some good assasins: Akali, Kassadin, Katarina, Poppy

Team Composition
Normally in NA Server we get: AP Carry, AD Carry, Support, Tank/Bruiser and Jungler/Tank Jungler, AP and Tank/Bruiser solo mid and top respectively, and the AD Carry goes with a support bot lane, this is mostly because AP Carries need mostly levels, and AD needs mostly items.
EU introduced an alternative composition which is sometimes used both in NA and EU, which is: AP Carry mid, AP Carry top, AD Carry + Support bot, Jungler (Sometimes tank + jungler), sometimes you get no tank though, this works rather nicely sometimes, against some comps.
You can of course tweak some things, but those are the most common compositions.

When playing ranked, most of the time you also want to counter pick, if they have Fiddles, you might want Janna, if they have Karthus, Soraka is a nice counter, but that itself is a quite wide topic which should be discussed by itself.