I've been playing Ezreal for about a week now, but I have no idea what a Carry does yet. Can someone please explain their role and what they excel at?
I'm not sure if this is relevant, but I always go AD, so I suppose I'm an AD Carry.

a carry is someone that has a very strong late game and if feed should be able to kill off most of the other team, if protected.

Carry is league of legend just means you are primarily a damage dealer. That's your role.

I know there is a cute saying that carries are champs that are weak early game but carry their team late game. But that really isn't true in LoL.

There are plenty of carries that are strong early and mid game. Ezreal has a really strong mid game.

Tristana, Graves, Cait, and Sivir can really mess somebody up early game if given the opportunity and right conditions.

There isn't a carry that is weak late game though...if built properly. But their strengths are not equal.

But what is consistent if that their row in a teamfight is to be the main source of damage. AP Carries are similar. They're mostly there for their damage output. Sure, some carries have great CC/initation but damage is really their primary role.

Well, AP Carries tend to be more bursty than AD Carries. AD Carries tend to rely on autoattacks late game and generally does more consistent damage. AP Carries tend to be bursty...even AP Carries like Ryze and Cass tend to have spikes in their damage output depending on their cooldowns.

O.o thnks guys . I just knw it. I really dnt knw all this time thanx

I'd say that the (obvious) difference between Ranged DPS (AD Carry) and Mage/Nuke (AP Carry) is the damage they deal.

AD carries generally build AD, attack speed, and crits for autoattack damage. Some deal pretty bursty damage. Short-term damage steroids like Graves's dash and Tristana's AS boost increase damage output for a short period.
They deal physical damage, and are countered by armor.

There's also melee carries, who are AD but don't fulfill the same role as a ranged carry. They can't assault a tower from safe range, and cannot consistently survive teamfights without completely destroying the opposition/bailing at the right moment. But they have more damage steroids, invincibility periods, or other lulzy stuff that lets them build damage without fear... if only for a while.

AP casters use abilities to deal damage. Most of them build AP, some get CDR. Many have stuns, silences, or other CC - Crowd Control, aka interrupting, spells. Many mages have strong ults as well as good AOE damage.
Almost all deal pure magic damage. Only exceptions are Lichbane (which deals physical damage scaling off AP) or Ahri's Orb, which deals true damage on the way back.

Burst damage is high - ult spells help in bursting down the enemy team before you die. Mages cannot 1v1 targets as easily if they've blown all spells in the fight, but they can stay back while cooldowns refresh.