From computer terminology for a trial version tested by users, it refers to a draft story that is being reviewed by a someone other than the author, usually referred to as a "Beta Reader." "Unbetaed" means that no one else has reviewed spelling, grammar, etc. "To beta" is the act of reviewing a story.
Like a proofreader or editor, these folks read a fanfic before it's posted. Some are good at catching spelling/ grammar/ continuity mistakes, and some are not. Usually they are just frequent visitors to forums without any credentials.
proof-reader/editor/first victim. the person who reads a story before it is posted on the net to stop it being posted as an incoherent mess of...well, messness. a beta is thus a good thing to have when you are making fic happen. make sure your beta can spell.