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Mary Sue is a term of approbrium slung at major characters.
Historically, it derives from an original character in original Star Trek fan fiction. The original Mary Sue was nowhere as bad a Mary Sue as many tagged with her name.
Mary Sues (and the male, known as Marty Stu, Gary Stu, or Larry Stu) are main characters who are often considered to be self-insertion alter-egos of the author. You have to be Mary Sue's author, or someone much like her, in order to enjoy the story.
Actually, this isn't too hard. As the poster child of Mary Sue success, we have the Anita Blake novels by Laurel K. Hamilton, down to Anita's looks being identical. (Undoubtedly, we all wish we had sales like this Mary Sue would rack up.) Mary Sue has also been called on Bella of the Twilight series.
A failed Mary Sue, on the other hand, is characterized by constant undeserved, unearned success. This annoys readers because most come to a book to struggle through difficulties to an earned success via their alter-ego of the main character. If you are considering actual professional publication, or even want good readership in fan fiction postings, you should try to lose the unearned aspects. On the other hand, if you're writing for your own good fun and Mary Sue gets you to 50,000 words--what can be wrong with her?