Actors Equity is the labor union for actors and stage managers in the United States. Founded in 1913 it is dedicated to fostering live theater in the U.S. The website is here.
The Catch-22 with Actor's Equity is that you can't get a job without the union card and you can't get a union card without a job. There are ways around this. Often actors can gain Equity points from appearing in some productions in regional theaters.
Due to rules regarding how Equity productions must be handled (frequent breaks must be provided. Food must be available. There's a minimum amount that they will be paid per production), many community theatres advertise for non-Equity actors and it's not uncommon for Alan Smithee to be credited for a performance due to an Equity actor wanting to avoid publicizing that he's working a non-Equity show.
See also Equity for Great Britain