AD - Attack Damage - The stat that directly affects auto-attack damage. It will also increase the damage of physical spells. Physical spells will have an orange (+ #) to show how much extra damage you get from your AD.
AP - Ability Power - This stat only works on abilities with a blue (+ #). The majority if not all mages use AP to increase their ability damage.
ArPen - Armor Penetration - Allows your auto-attacks and physical damage spells to ignore some amount of the enemy's armor.
AS - Attack Speed - The speed in which you auto attack (no effect on spells). An attack speed of 1.0 allows you to attack once every second, and is capped at 2.5 attacks per second.
B - Back - To signify that you are going to recall back to your base or otherwise retreat; named either for the word "back" or for the fact that by default, pressing b initiates recall.
BN - Baron Nashor - The boss monster which spawns near top-lane on Summoner's Rift, first at 20 minutes. Killing this monster grants the team-wide buff, Hand of Baron, which buffs all nearby minions attacks as well as defenses. It also grants bonus AP and AD, as well as an enhanced recall (B).
Blue - Blue buff, or the Crest of Insight, on Summoner's Rift. It is received by killing the Blue Sentinel, which appears in the eastern and western jungle. Provides 10% cool-down reduction (CDR) and increased mana regen.
Bot - 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 - Champions, usually ranged AD heroes, or high-damage mages. They are capable of dealing massive damage and taking down the enemy team, but have no way to defend themselves (they "carry" the entire team for the damage).
CC - Crowd Control - Abilities and effects that restrict a Champions ability to act or move, such as slows and stuns.
CDR - Cool-down Reduction - Reduces the cool-down time on your spells up to a maximum of 40%.
CS - 1. Creep Score; 2. Crystal Scar - 1. The number of minions you've killed. This is also increased when you benefit from an ally's Relic Shield, Targon's Brace, or Face of the Mountain. 2. The 5v5 map on which Dominion is played.
Dive - Also known as tower diving, it's the act of attacking an enemy under their tower, usually to deal a killing blow. 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 many abilities, such as Cassiopeia's Noxious Blast, the Crest of Cinders (red buff), or the item Liandry's Torment.
Drag - Dragon/Drake - The boss monster near bottom lane on Summoner's Rift. Each Dragon kill achieved by your team grants a stack of Dragon Slayer which gives buffs depending on the amount of stacks you have. See also: pull.
Gank - To flank an enemy champion by surprise and attempt to kill them. Usually done by the jungler, or a roaming mid laner.
Gp10 - Gold per 10 seconds - How much gold you generate every 10 seconds, by items, runes or masteries. Useful for support champions, since they are the only ones who are not getting gold from killing minions.
Hard Leash (red/blue) - Same as a normal Leash (see below), but leaving the monster with very little health (around ~10%) so the jungler can kill the monster without using Smite (usually to save it for the other buff). This has become less common since Season 4, due to the lower cool-down on Smite.
IMO - In my opinion - This is a common abbreviation across the web, and isn't limited to online games. Can also be IMHO for in my humble/honest opinion.
KS - Kill Steal/Secure - When someone finishes off an enemy that another teammate was attempting to kill. Whether it's considered a steal or a secure depends on the situation.
Kite - Kiting is basically "Killing In Transit." It involves running away or around while still dealing damage to your pursuer. It is done to avoid a straight up fight that you will probably lose, or to stay out of range of the enemies abilities and attacks.
Leash (red/blue) - Helping the jungler with a neutral monster until the monster is at ~30-40% of health, so that the jungler can use Smite to finish it. Those that are leashing will leave before the monsters are killed, this is so all EXP goes to the jungler.
MIA - Missing In Action - Signals that your lane opponent (or the specific enemy you mention) is no longer visible on the map, and may be on their way to "gank" a teammate. Everyone should be careful and watch out for the missing champion. Also used: "miss" or "ss"
Mid - The middle lane in Summoner's Rift, or the middle level of capture nodes in Dominion.
MM - MatchMaking - The System in LoL that puts your team together and then finds another team for yours to play against.
Mp5 - Mana per 5 Seconds - The stat which shows the total mana your champion recovers within 5 seconds.
MPen - Magic Penetration - Allows your magic damage abilities to ignore some amount of the enemy's magic resistance when they hit.
MR - Magic Resistance - Allows your champion to reduce the incoming magic damage done by enemy abilities.
Poke - Harassing the enemy team with long range abilities like varus' piercing arrow and similar skills. This is done when the two teams are facing off before a team fight or when contesting an objective, in order to put the other team at a disadvantage when the fight starts.
Proc - Also known as on-hit effects. These can come from items, masteries and most often champion abilities. Examples include the passive Thunderlord's Decree (basic attack hits), the item Phage's move speed buff (basic attack hits), Vel'Koz' passive damage (ability hits) and Vi's Denting Blows (both basic attacks and ability hits).
Pull (red/blue/dragon) - Pulling no longer works on red and blue buffs as of season 3. Pulling refers to attacking dragon and "pulling" it out of the pit. This makes it harder for the enemy team to jump over the wall and smite it, and easier for your team to leave the area since you won't be clustered in the pit.
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, or the Crest of Cinders, on Summoner's Rift. This is received by killing the Brambleback in the northern and southern jungles. The buff adds an on-hit effect to auto-attacks which damages and slows based on your level. This affects all champions, monsters and even towers. It also gives you a bonus to health regen.
RH - Rift Herald - A boss monster introduced in (pre)season 6, who appears in the Baron Nashor's pit prior to his own spawn at 20 minutes. Gives the buff Doom's Eve to killer which buffs the attacks of surrounding minions, as well as increased damage and move-speed.
SR - Summoners Rift - The most popular map, classic 5v5. Has 3 lanes, and the Dragon, Rift Herald and Baron Nashor bosses. Also: "Rift"
SS - 1. Sight Stone 2. See MIA above - Sight Stone is an item which gives you 4 wards which are recharged when returning to base. Picked up most often by supports and occasionally junglers.
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, leading from the river into their respective jungles and lanes.
TT - Twisted Treeline - The smaller map for 6 players, has 2 lanes.
TP - Teleport - Summoner spell used to get from one location to another after a short channel time. Can be cast on allied minions, wards and structures.