AA: Auto-Attack. A term for champions' standard or basic attack animation command.
AD: Attack Damage. Physical damage from your auto-attacks or from spells that deals physical damage before applying the targets' armor, they are usually displayed as an orange number in the champions' abilities' window and are shown as a red number when shown damage dealt to targets in combat.
AoE: Area of Effect. Spells that usually can hit multiple targets at once.
AP: Ability Power. Magical damage from spells and maybe from some auto-attacks (depends on masteries and items) before applying the targets' magic resist. They are usually displayed as a green number in champions' abilities' window and are shown as a purple number when shown damage dealt to targets in combat.
AP/Lvl: scaling Ability Power per Level. The amount of ability power you gain from runes and masteries based on your champion's level.
AS: Attack Speed. The amount of auto-attacks' animations your champion can execute within 1 second. Or how fast and how many times your champion can auto-attack within that same time frame.
CC: Crowd Control. Negative status effects that can disrupt you from using certain commands and actions or disable you completely.
CS: Creep Score. The amount of minions and monsters you've slain or have last-hitted successfully.
DoT: Damage over Time. The amount of damage a spell or attack can do over a certain period of time.
GP10: Gold income generated Per 10 seconds. The amount of additional gold income that can be attained every 10 seconds to help fund you.
Mana Regen: Mana Regeneration. A champion stats that displays how much mana a champion can regenerate every 5 seconds.
MPen: Magic Penetration. Flat Magic Penetration (from runes and certain items) ignore a certain flat amount of the target's magic resist, while % Magic Penetration (from masteries and Void Staff) ignores a certain percentage of the target's magic resist.
MR: Magic Resist. A champion stats that affects how much magic damage a champion will receive. The more magic resist a champion has, the less incoming magic damage they will receive to their health.
MS: Movement Speed. A champion stats that displays how quickly your champion can move across from one location to another location on the map. The amount of units it can travel within a certain time frame.