A beta reader is a proofreader for one's fanfiction. It is important to note that a beta, from the second character in the Greek alphabet, is the second pair of eyes needed for a fanfic: the first eyes you need are your own. The beta's job is to point out any problems with flow, plot consistency, characterization, spelling, punctuation, and grammar of the story. Your job is to then fix those problems.
It is a good idea for every fanwriter to get their work reviewed by a beta. When writers claim to have one when their fics are riddled with errors, it is insulting to their readers, themselves, and their betas - if they do actually have them, which is doubtful.
A good beta reader should be treasured. That means taking their advice, or at least giving it some very careful thought before deciding not to take it. Showering them with gratitude and praise, doesn't hurt either.