Hey there, I will try and help,
PADI- Professional Association of Dive Instructors
NAUI- National Association of Underwater Instructors
SSI- Scuba Schools International
NACD- National Association of Cave Diving
IANTD- International Association of Nitrox and Technical Divers
NSS-CDS- National Speological Society-Cave Diving Section
GUE-Global Underwater Explorers
DIR- Doing It Right(a way of diving, taught by GUE)
OW-Open Water(your first certification)
AOW- Advanced Open Water(usually the next certification after OW)
DM- Dive Master(usually the first step to being a professional)
AI- Assistant Instructor(second step to being a professional)
OWSI- Open Water Scuba Instructor
IDC- Instructor Development Course(What you must pass to become an instructor)
IE- Instructor Exam
BCD or just BC- Buoyancy Compensating Device or just Buoyancy Compensator
Reg- usually refers to your regulator or the regulator set as a whole
BP/W- Back Plate and Wings(a type of BCD used for technical diving)
Tech- Technical diving
Hog- Hogarthian (a type and/or style of diving, used by tech divers usually)
STA- Single Tank Adapter (device used to convert a doubles setup to a single tank)
doubles- usually having two tanks hooked together with a manifold to double your air supply
D-ring- usually used on your BCD to hook things to, mainly used in tech diving
Stage- extra tank taken with you to extend your dive time, used for tech diving
Deco- decompression, again, tech diving
NDL- No Decompression Limits
SMB- Surface Marker Buoy
I am sure I will think of more later, that and a bunch or other people will probably chime in with some that I missed. Should help some. Mark