What are the common abbreviations in crochet patterns and more importantly what do they mean?
There are about nine or ten main stitches that are the most commonly used and, of them, four or five are probably used the most.
Most common US stitch abbreviations:

ch - chain
bpsc (or bpdc, bptc) - back post single crochet (double crochet, triple crochet)
dc - double crochet
dtr - double treble crochet
hdc - half double crochet
fpsc (or fpdc, fptc) - front post single crochet (double crochet, triple crochet)
sc - single crochet
sl st or ss - slip stitch
tc - treble crochet
trtc - triple treble crochet

Most common UK stitch abbreviations:

ch - chain
dc - double crochet
dtr - double treble crochet
htr - half treble crochet
rtrb - raised treble back
rtrf - raised treble front
ss - slip stitch
tr - treble crochet

Other standard, non-stitch related abbreviations:
These may be used on their own or in conjunction with the abbreviations above.

bp - back post
dec - decrease
fp - front post
inc - increase
rep - repeat
rnd(s) - round(s)
RS - right side
st - stitch
tog - together
WS - wrong side
yo/yoh - yarn over/yarn over hook

Also, many patterns use an asterisk (*) to mark a repeated section of work.
There are other terms that some sites may define like pc (popcorn) or cl (cluster) but these should be defined in your pattern because what each pattern considers a popcorn stitch or cluster will vary greatly. They are non-standard, so look for a section in the instructsion that breaks up these terms into their smaller parts for you.