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 - Creep Score - the count of the number of creeps you've killed. Higher number means you've been better at last hitting, and have more gold to spend.
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.
Drag - Dragon - the boss creep near bottom lane on Summoner's Rift.
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).
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, usually but not always based on % chance. Examples include Phage's Slow, Vi's Denting Blows, Caitlyns Headshot, etc.
Pull (red/blue/drake) - 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.
Red - Red buff on Summoner's Rift, from the Lizard Elder. Scaling Damage over Time and Slow.
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.
TT - Twisted Treeline (3v3 map) - The smaller map for 6 players, has 2 lanes.