I honestly wish I'd kept a copy of the first essay I read about Mary Sueism. It was well-done, and I don't think I can do the subject half the justice the original did. However, I'll give it my best shot.

You already know Mary Sue. Mary Sue is the perky, bright, helpful sixteen-year-old ensign who beams about the ship. Everyone on the ship likes Mary Sue, because Mary Sue is good at everything. Mary Sue is an engineer, a doctor in training, a good leader, an excellent cook, and is usually a beautiful singer. Mary Sue often has mental powers that may manifest themselves as telepathy, precognition, or magic. Her past is tragic, more so than any other character on the series. (Many Mary Sues have a backstory that reads like a V.C. Andrews novel. This is a clue.) If Mary Sue is very young, she is often the offspring of one or two already established characters. If she's a little older, she will probably end up sleeping with the author's favorite character. Sometimes, she fills both roles. Her name is often the author's name, be it a net.

name, a favored nickname, or the author's middle name (this is seen in the most famous Mary Sue of all time, Wesley Crusher, who was named after Trek creator Eugene Wesley Roddenbery). By the end of the story, Mary Sue will be in bed with the desired character, will have beamed away amid cheers from all the regulars, or will be dead, usually accompanied by heavy mourning from the cast. The reader, on the other hand, will be celebrating. BTW, Mary Sue's twin brother can often be identified by his brooding, solitary behaviour, matched by his maverick disregard for authority (for a great example, see the very beginning of TNG's "Hollow Pursuits" alias Barclay, Part One).

Before I go any further, I would like to point out that I have read several excellent stories with characters that fit every part of this description. Fortunately, there are authors who can take this character type, and make a figure just as memorable and vibrant as any ever seen on the series. For the most part, unless you know what you're doing, if you see a character of your own fitting this description, find another way of telling the story. Please. Remember, fanfic is (for many of us) about characters we know and love, not about how much they like a new person who has nothing to do with their universe. Mary Sues not only stop fanmail, they often invite flames. Use them wisely, or not at all.