I live in Raleigh.
There is the APA, but frankly I avoid it. Don't know if it's like this everywhere, but my experience with the APA around here was lot's of politics, cliques, conflicts of interest, and the sandbagging/handicap "management" was totally out of control.
There is also a new league called NAPA, which is just starting out but has been fun so far.
There are quite a few tournaments around here. Mostly 9-ball. I shoot at Brown's Billiards and Brass Tap. Both good halls with decent, full-size tables. There is also Sharky's, New York Billiards and many others, but I don't know much about them. Pool is HUGE in North Carolina.