VixMom:
Actors Equity Association is the professional union for stage actors, singers, dancers and stage managers. There are many ways to get into the union - none of them overly easy. The "easiest" way to get in is to simply be hired into an Equity contract. Another way is to work your way through in Equity theatres in what's called the Equity Membership Candidacy program. As a non-Equity actor, an EMC has to work 50 weeks in an Equity theatre (usually in minor supporting or ensemble roles). After 50 weeks you can pay your dues and join up. After a certain period of time, members of "sister" unions may also join AEA.
Once you're a member of Equity, you're a member - just so long as you pay your dues. You're not permitted to perform in non-Equity roles, though many people scab if just to get some work. Sadly, approximately 98% of Equity stage actors are unemployed in their chosen field, and many choose to stay in the union even after giving up "the biz" because it's such an accomplishment to get in.
There are many other performance unions, including AFTRA (for film, radio and TV), AGVA (American Guild of Variety Artists [generally ballet, jazz & other professional dancers]), SAG (Screen Actors Guild [for TV & film]), AGMA (opera and orchestras), AFM (pit orchestras & recording musicians), SSDC (Society for Stage Directors and Choreographers), and IATSE (designers, crew, stagehands & front-of-house).
Loads of unions. And I'm not in any of them.
Yet.

DRs Ben, Jose, Elmore, and BK can certainly fill you in on the various unions. I think others on the board can, too, though I can't remember who's union and who isn't.