AD - Attack Damage - The stat that directly affects melee auto-attack damage. Many melee/fighter/tank champions will want to get AD gear to increase their damage.
AP - Ability Power - The stat that directly affects ability damage. Many mage/ranged champions will want to get AP gear to increase their damage.
ArPen - Armor Penetration - makes your AD attacks ignore some amount of the enemy's armor when they hit.
AS - Attack Speed - The speed in which you attack with normal attacks (not spells). AD champions benefit from this.
b - 1. Back; 2. Baron - 1. To signify that you are going to recall into your base or otherwise retreat; named either for the word "back" or for the fact that by default, pressing b initiates recall. 2. Refers to Baron Nashor, the boss monster who spawns at 15 minutes and is found near the top lane on Summoner's Rift.
Blue - Blue buff on Summoner's Rift, from the Ancient Golem. Provides scaling CDR and bonus mana/energy regen.
Bot - generally refers to the bottom lane in Summoner's Rift and Twisted Treeline, or the champions who are laning there. E.g., "Bot mia" = "the enemy champions in Bot lane are missing, beware." On Dominion, this refers to the lower-left or lower-right capture points, as well as the bottom lane in general.
Carry - heroes, usually AD heroes, capable of dealing massive damage and taking down entire groups, but usually die themselves easily too (they "carry" the entire team for the damage)
CC - Crowd Control, which refers to abilities that restrict a Champions ability to act or move, such as slows and stuns.
CDR - Cooldown Reduction - The more %, the quicker the cooldown on spells is over (a maximum of 40%). Usually good for casters or heroes without mana.
CS - 1. Creep Score; 2. Crystal Scar - 1. The count of the number of minions you've killed. This count is also increased when you benefit from an ally's Relic Shield, Targon's Brace, or Face of the Mountain. Higher number means you've been better at last hitting, and have more gold to spend. 2. The 5v5 map on which Dominion is played.
Dive - Tower diving, the fact of attacking an enemy under his tower, usually to deal a killing blow on a low-health champion. Doing so makes the tower attack the diver, making it a risky move.
DoT - Damage over Time - An effect whereby a player or monster loses health over time. DoTs may be applied by abilities, such as Tristana's Explosive Shot, by buffs, such as Crest of the Elder Lizard (red buff), or items, such as Blackfire Torch.
Drag - Dragon/Drake - the boss monster near bottom lane on Summoner's Rift. See also: pull.
Gank - To gang up on an enemy champion by surprise and kill him/her.
Gp10 - Gold per 10 seconds. How much gold you generate per 10 seconds, by items, runes or masteries. Useful for support champions.
Hard Leash (red/blue) - same as a normal Leash (see below), but leaving the monster with very little health (around ~10%) and the jungler finishes the monster without using Smite (usually to save it for the other buff since he won't be receiving help this time). This is less common in Season 4, due to the lower cooldown on Smite.
IMO - In my opinion, sometimes IMHO - In my humble opinion. This is a common abbreviation across the web, and isn't limited to online games.
KS - Kill Steal - When someone finishes an enemy, but other teammate wanted to perform.
Leash (red/blue) - helping the jungler with a neutral monster until the monster is at ~30-40% of health at the beginning of a game, so that the jungler can use Smite to finish it (usually people leave the area so that the jungler gets all the experience).
mia - Missing In Action - Means that the enemy in your lane (or the specific enemy you mention) is no longer visible in your lane, and may be on his way to "gank" a teammate. Everyone should be careful and watch out for the missing champion. Also used: "miss" or "ss"
Mid - refers to mid-lane in Summoner's Rift, or the middle level of capture nodes in Dominion.
MM - MatchMaking - The System in LoL that teams up people/puts them against each other.
Mp5 (Aka MP/5) - Mana per 5 Seconds - a stat referring to how much mana your champion recovers every 5 seconds.
MPen - Magic Penetration - makes your magical attacks ignore some amount of the enemy's magic resistance when they hit.
Proc - Refers to an item's or an abilities' activation due to some other action, usually but not always a basic attack. Examples include Phage's move speed buff (basic attack), Vel'Koz' passive (spell damage), Vi's Denting Blows (both basic attacks and spell damage), Caitlyn's Headshot, etc.
Pull (red/blue/dragon) - the fact of hitting a neutral monster, usually at range, so that it will follow you for a few seconds, allowing another champion to attack it without getting damage. You will find this typically at the beginning of the game, with the carry pulling the golem/lizard for the jungler. As of Season 3, this no longer works on red or blue.
OOM - Out of mana; a champion who is out of mana usually has to retreat and will be less helpful in a fight.
Red - Red buff on Summoner's Rift, from the Lizard Elder. The buff adds a DoT and slow on-hit effect to auto-attacks.
SR - Summoners Rift (5v5 map) - The most played map, the classic DotA 5v5. Has 3 lanes, the dragon mob and the boss Nashor. Also: "Rift"
Top - Top lane in Summoner's Rift/Twisted Treeline, or the upper-left/upper-right capture points in Dominion.
Tri - The bush shaped like a three-pointed star. Found in the bottom lane on Blue (Green) side, or the top lane on Red (Purple) side.
TT - Twisted Treeline (3v3 map) - The smaller map for 6 players, has 2 lanes.
TP - Teleport - Summoner's spell