There's about as many different definitions of "Mary Sue" as there are people who talk about it, so I'm going to write my own:
A Mary Sue is a wish-fulfilment character, who is absurdly competent at anything they do, and/or loved by everyone else, and/or repeatedly saves the day, and/or steals the show from what should theoretically be the protagonist, all with no good (or none at all) in-story explanation.
The name comes from a parody story published in 1973 in a Star Trek fanzine (see page 25 of this history of ST fanzines for the full text), but soon spread to many other fandoms. One of the most problematic aspects of the term is that it's a woman's name, and many people can't seem to conceive of a badly-written, over-powerful wish-fulfilment character that also happens to be a man: consider that, after all, Batman has most of the traits of a Mary Sue, but very few people seem to notice.
For a very in-depth, but not exactly balanced, analysis of the idea, TVTropes is always a good starting point, and the Wikipedia article has useful references.
