Proc - Chance for a special ability to happen. Example=Plasma DoT from plasma weapons.
Mob - Classically Monster or Beast, refers to any enemy controlled by the computer.
QQ - Looks like crying eyes, used to cry and/or complain.
CC - Crowd Control. Example=Viral Matrix
DoT - Damage over Time
HoT- Heal over Time
Here are some others you will probably see:
LFG/LFT - Looking for Group/Team
LFM - Looking for More, for a group
BoE - Bind on Equip, item will bind to you(untradeable, but can sell to vendor) when equipped
- Also, Badge of Exploration. Currency used in Sol to buy equipement, reward from explore missions
BoP - Bind on Pickup, item will bind to you when you pick it up
- Bird of Prey, Klingon warship
DSE - Deep Space Encounter, any random encounter on the sector map
AoE - Area of Effect, any ability which affects an area instead of a single target
PBAoE - Point Blank AoE, centered on the person using the ability
Hope that gets you started, and welcome to the world of MMOs