Champions are the characters you can choose from to play as in your team .
A game of League of Legends can either be on 3's or 5's (Twisted Treeline or Summoners Rift)
Your aim of the game is to passthrough the enemies' defenses (3 Turrets->Inhibitor(doesn't attack back)->2 Nexus Towers; this is in each of 3 lanes: top, middle and bottom lanes, however you don't have to kill every single tower to finish the game but it makes it alot easier to finish the game if you did) to finally kill their Nexus to win the game.
Along the way you'll obviously encounter enemy champions (real players), minions (the things you have to farm (kill) to gain money to buy items to become powerful and other places of struggle that you find both teams fighting over. (Baron and Dragon) Killing baron gives you a buff which gives you a boost in all stats for 4 minutes and killing Dragon gives 200 ish gold to everyone on your team.
The meta-game for season 3 is the same as season 2: Bottom lane has a: Support which buys aura items, wards, and generally has shielding skills or healing skills and a AD Carry or Attack Damage Carry which usually has a long range, fast attack speed and alot of damage but are very, very squishy (easy to kill)
Middle usually has an AP carry there or AP Tank Carry (Ability Power Carry i.e. Magic user) this champion usually has VERY HIGH BURST DAMAGE or ALOT OF STUNS/SNARES/ROOTS etc to keep enemies immobile whilst the AD carry, jungler and top laner dish out some damage. But not all the time. Sometimes you can have an anti-AP Carry to counter those annoying champions that counter most AP's. Such as Talon and Zed, they're both slipper AD casters that can dodge alot of skills and have high burst-abilities.
The jungler is the person who roams around the jungle killing neutral monsters to free up a space top lane so that top laner levels up faster than bottom and you'll have 3 high leveled champions on your team instead of 1. There are many different types of junglers. Counter-junglers (they focus on stealing the enemies jungle), fast clearers (they clear the jungle faster than most and gank afterwards), gank based junglers (once they get all the necessary skills to pull of a good gank *when someone on your team or enemy team comes to help in the killing of the enemy* they go to a lane where they think it will help the most) the jungler can either be a tanky jungler (slow with alot of cc and cc reduction) cc = crowd control (stuns/snares/silences/roots/knockups etc...) or an off-tank (has some defenses but still has alot of damage) it's best to have either a tank and an off tank on your team, but you can have 2 off tanks and a tank if your support is playing Taric, Alistar or Nautilus.
Top: Top's usually are off-tanks but you can play tanks at top against other tanks or against champions with low damage starts otherwise you will lose your lane.
That's still not enough to explain this game if you're confused about anything else just ask me lol. I've been playing since closed beta.