Basic Trope: An Author's pet character created by bad writing, who bends the universe of the story around him or her.
Straight: In a "Bob The Strong" fanfic, Bob rescues Princess Mary Sue from the Dragon in the first chapter. The sparkly maiden refuses to go back to her castle and follows him instead. In chapter 2, she all of a sudden becomes a strong and powerful character without any reason. All the good characters love her and the evil characters hate her. Even though she's a total bitch the narrator tells us she's pure. Within a month of meeting her, Bob dumps Alice to be with his one true love (guess who?)
The canon heroes never call her out on this and instead worship her for this saying it was the right thing to do. The one character who doesn't is looked down upon by the others until they, eventually comes around and worship Mary Sue like everyone else. She then takes control of the empire and uses it to found Mary Suetopia and rule it as a goddess forever. See also: Thirty Sue Pileup
Downplayed: Sue is great at a lot of things and is the most well rounded, emotionally stable and mentally functioning member of the cast, but they're mostly a supporting character a best, whose main contribution is the occasional Rousing Speech or telling The Hero an old tale from their many travels that helps them accomplish a certain task, serving the plot mostly by being The Heart or The Ace.
The character discovered the secret to perfection.
The character is a Reality Warper who modes and shapes reality to its/her/his liking.
A character with few-to-no positive traits, intended to be as unlikable and as much of a failure as possible, who nobody likes.
Marietta appears in the fan fic intended as comic relief, including her infatuation with Bob the Strong. All the readers find her funny, charming, and intelligent and think that without author manipulation there's no way that Bob would not have seen that she's much better for him than the vain and lazy Alice — but the author does not perturb the canon pairing, even though Bob has no reason not to prefer the better woman.
Mary Sue becomes the centre of attention, and hates the amount of attention she gets, and every time she tries to toss the spotlight to someone else, it's thrown back, eventually driving her crazy because she hates being the centre of attention so much.
Because no-one else has the skills she does, Mary Sue is forced to do unpleasant things.