This will help a bit: http://www.lolking.net/champions/
second column from the right, "Meta"
EDIT: Alternatively, use the same link, enter champion name in the search bar to the top, then click "Guides" in the box that pops up. Then, it tells you the role on the left of each guide, and with this you can check if champions have multiple metas, such as Teemo, where you find mostly top guides, but some mid and one jungle (as opposed to Ryze, where you find pretty much only mid guides)
EDIT: Just realized my answer probably won't make much sense to you. In the current meta, more popular type first, this is what the general rule is:
top = AD bruiser/melee, AP carry
mid = AP carry
bot = support+ranged AD carry
jungle = bruiser, assassin, or misc. Tends to be champions with good sustain or good ganking power. Tends to be tanky.
For bottom, you can click on a guide, and if you see heavy Offensive masteries, it's probably AD carry; if you see more Utility masteries, it's probably support. Looking at itemization in the guide, AD items like Infinity Edge, Bloodthirster, Statikk Shiv, or Phantom Dancer are dead giveaways that it's an AD carry. For support champions, you'll find things like Philosopher's Stone and Aegis of the Legion.
The best way is probably to just learn and memorize.
EDITS FOR GREAT JUSTICE: Realized lolking is smart and does differentiate between AD Carry and Support champs. So you don't need that part about clicking on a guide. Eh. You can still use the Guides method to find out if a top is AD, AP or hybrid - I just checked Kha'Zix since I know he's being played mid with new patches, and I found out he's AD, because I see AD Offensive masteries advised (armor penetration). Generally if a champ can be both mid and top though, it'll probably be APC.
If you know a champion is mage-like/builds AP, it's probably a mid.