There are a lot of common abbreviations and terms in "Geocaching Lingo" - I'll cover the most common ones here, but if I've missed any, just ask!
Cache = Geocache
Cacher = Geocacher
Caching = Geocaching
Ground Zero = Immediate area where the geocache is hidden
Sig Item = Signature item - sometimes cachers will buy personlized items like pencils, carribeaners, wooden nickels, etc. with their geocaching name on them that they use as trade items
Log = Log sheet or log book
Muggle = Non-geocacher
Coin = Usually refers to a geocoin
Multi = A multi cache - a cache with smaller caches (stages) that provide information to find the final geocache
Swag = Items for trading
TB = Travel bug
TNLNSL = "Took Nothing Left Nothing Signed Log"
TFTC = "Thanks For The Cache"
FTF = First to find
STF = Second to find
DNF = The dreaded sad-face - "Did Not Find"
If I've missed any let me know and if I know it, I'll add it. :-)
Here are some more abbreviations (suggested by Kevin - thanks!):
GZ - Ground Zero
TFTH - Thanks for the Hide (a derivative of TFTC)
GPSr - Global Positioning System receiver
PnG (P&G) - Park and Grab
CnD (C&D) - Cash and Dash
CITO - Cache In, Trash Out - Refers to an ethic whereby cachers try and leave an area in better shape than they found it by picking up any trash they encounter after finding a cache.