David Hammond, Flux, and Halcyan 2 contributed.

This is a quick guide to the V:TES specific abbreviations:

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Disciplines:

ani / ANI - Animalism (inferior ability / superior ability)
aus / AUS - Auspex (inferior ability / superior ability)
cel / CEL - Celerity (inferior ability / superior ability)
chi / CHI - Chimestry (inferior ability / superior ability)
dai / DAI - Daimoinon (inferior ability / superior ability)
dem / DEM - Dementation (inferior ability / superior ability)
dom / DOM - Dominate (inferior ability / superior ability)
for / FOR - Fortitude (inferior ability / superior ability)
mel / MEL - Melpominee (inferior ability / superior ability)
myth / MYTH - Mytherceria (inferior ability / superior ability)
nec / NEC - Necromancy (inferior ability / superior ability)
obe / OBE - Obeah (inferior ability / superior ability)
obf / OBF - Obfuscate (inferior ability / superior ability)
obt / OBT - Obtenebration (inferior ability / superior ability)
pot / POT - Potence (inferior ability / superior ability)
pre / PRE - Presence (inferior ability / superior ability)
pro / PRO - Protean (inferior ability / superior ability)
qui / QUI - Quietus (inferior ability / superior ability)
san / SAN - Sanguinus (inferior ability / superior ability)
ser / SER - Serpentis (inferior ability / superior ability)
spi / SPI - Spiritus (inferior ability / superior ability)
thau / THAU - Thaumaturgy (inferior ability / superior ability)
than / THAN - Thanatosis (inferior ability / superior ability)
val / VAL - Valeren (inferior ability / superior ability)
vic / VIC - Vicissitude (inferior ability / superior ability)
vis / VIS - Visceratika (inferior ability / superior ability)

Note that these are typically used, but some similar variants are used
too - e.g. THA for superior Thaumaturgy.


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Clans:

Typically, clans are typed by name (e.g. Brujah). However, for Sabbat
clans who have a Camerilla or "normal" equivalent, they are typed with
a "!" in fromt to denote Antitribu. e.g. !Brujah = Brujah Antitribu.

Most common abreviations:

Followers of Set - FoS
Malkavian - Malk
Nosferatu - Nos
True Brujah - Trujah
Blood Brothers - BB
Daughters of Cacophony - DoC

HoS - Harbringers of Skulls 
Cats - Ahrimanes
Nossies (as well as Nos) - Nosferatu

And as a rule of thumb, the first three letters of ANY clan are
sometimes used to abbr. e.g. Bru = Brujah.


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General:

S:CE = Strike: Combat Ends. 

OOT = Out of turn (Master) 

Rush = directed attack (typically Bums Rush, but the term is given to
encapsulate most ways of getting into combat with another minion).

S+B = Sneak Bleed

NRA = No Repeat Actions - a rule that prior to the last few months was
just a tournament floor rule.

CL = Card Limits. n.b. Some people play with card limits as a house
rule for deck construction. Others don't. This tends to cause all
sorts of arguments on the newsgroups. If you see a post with anything
to do with "CL" or "Card Limits" in the title, avoid (unless you
really want to join in the scrap!).


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Sets:

CE = Camerilla Edition

AH = Ancient Hearts

DS = Dark Sovereigns

FN = Final Nights

SW = Sabbat War

BL = Bloodlines
 
n.b. Jyhad, VTES, and Sabbat, tend to be said in full.


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Newsgroup:

OT = Off Topic

LSJ = L Scott Johnson (netrep, lead designer, all round dude 
                        - the no. 1 "V:TES Celebrity")


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Here's some for specific cards:

BO = Bewitching Oration
DI = Direct Intervention
DotB = Drawing out the Beast
FSR = Free States Rant
HG = Hunting Ground
IG = Immortal Grapple
KRC = Kine Resources Contested
MEE = My Enemy's Enemy
MT = Minion Tap
OOT = Out-Of-Turn (Master)
PB = Powerbase
PC = Perfect Clarity
PS = Praxis Seizure
PTO = Protect Thine Own
RtI = Return to Innocence
SR = Sudden Reversal
TGB = Toreador Grand Ball
TitR = Tension in the Ranks
ToGP = Temptation of Greater Power
ToV = Taste of Vitae
TS = Torn Signpost
TT = Telepathic Tracking
TVC = Telepathic Vote Counting
Queen of Sleeze = Arika


Of course any card could probably be referred to by its initials, but these are
the much more common ones.


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There are also some other common terms. Some use "governing down/up" or
something similarly for the superior of Govern the Unaligned (and Scouting
Mission and Enchant Kindred). Some use the term "bounce" for Deflection and
Telepathic Misdirection (the latter is sometimes referred to as "the finger" or
something like that in some groups due to the artwork of the CE version). Etc.

Wakes. "Wakes" generally refers to any means of untapping a minion or
let it act as if untapped when not in your turn. Typically, it refers
to reaction cards that let you untap, like Wake with Evenings
Freshness - from when the term comes from - Forced Awakening, Guard
Dogs, Rat's Warning, etc. However, there are some permenant cards that
allow this mechanic (most notably, Eternal Vigilance).