When you do something onstage, you do it in front of an audience. Onstage tap dancing can be exhilarating to watch if the dancers are good, or embarrassing if they're not.
An onstage mistake, like a forgotten line or a dropped prop, can sometimes be concealed by quick-thinking actors. Your role in the school play might turn out not to have any lines at all, but to involve your character lurking around onstage looking mysterious. The opposite of onstage is offstage, meaning things that occur where the audience can't see them.