Basically, think non-AEA ASM . . . but the duties are SO flexible, that it may change from project to project in a theater and vary from theater to theater. Right now, I have a PA running the deck as ASM on one show, another PA is in the rehearsal room tracking costumes and being on book. Sometimes, the PA is only dealing with paperwork outside rehearsal, other times that are a vital park of the SM team.
It really depends on the needs of the show and the stage management team.