Before I begin I will admit there is significant debate over what is and is not a Mary Sue. About who is and who is not, and about how best to explain it. Let me jump over all that and specify a kind of Mary Sue; perhaps the worst kind, an unbelievable character.
Fiction isn't meant to correspond to reality. It is only by our willingness to believe what is happening that a story proceeds. We take the events of Harry Potter on faith. It “happened” because the author, Joanne K. Rowling, said so. Like a train, the story chugs along, and impossible things occur one after another. We believe these, we suspend disbelief and trust in her, and our own imagination.
This trust is broken by the introduction of something unbelievable. If, for example, Santa Claus showed up in a Harry Potter book. Santa Claus pulls his reindeer, Rudolf, out of his magic sack and threw it in front of a curse to save Dumbledore. If this happened, we would have some questions. We might believe in Santa Claus. We might enjoy reading Harry Potter, but there is something unbelievable about Santa making an appearance at Hogwarts.
The greatest problem with this kind of unbelievable people and occurrences is that it is easy to do. It really isn't that hard to think of our fictional heroes as larger than life, or even godlike. But, we can't put these impossible people into a story they are mismatched for.
Consider with me the worlds smartest woman. Done, we have imagined her. Now imagine a movie where the world's smartest woman falls in love with the pastry chef down the street. There is some mismatch here, this pairing begs too many questions. A romance movie where a woman falls in love with a neighbor is believable, and has been done before. But, it doesn't require and is hampered by larger than life detail. What does she see in the chef? how did they meet in a meaningful way? Why do we believe she is the smartest woman in the world? More importantly, how does it serve the fiction to be the smartest?
We might ignore these things in a different story, the same way we might ignore the detail that Santa always wears red. But, this detail in the romance is out of place. Characters need context. Fiction can be anything, but that doesn't mean it should. We may want to make our heroes gods. But godly characters need god like responsibilities.
To reinforce this idea, that an extraordinary character needs to be in an extraordinary story; lets imagine a commonly accused Mary Sue, Superman. Superman is clearly impossible, he is purposefully perfect. One of his more unbelievable cohorts is Jimmy Olson.
Superman can bend steel, can out-run a rocket, heck, he can fly! Jimmy takes pictures for a newspaper. There is a problem with this pairing. Jimmy Olson is described as “Superman's Pal,” but the idea that Superman would elect to take Jimmy on an adventure with him is ridiculous. Jimmy isn't prepared to accomplish even 1% of the things Superman can. Jimmy should get killed in any issue when the man of Steel decides to confront armed gun men with his pal in tow. If jimmy Olson survives every encounter with horrible monsters, and giant robots, we beg some questions. How is he so lucky? When does he do his job? How does he get home after being flown to the top of a building for his own safety?
If I wanted to put Superman in a story with Jimmy Olson, I should focus on the mismatch. Superman doesn't schedule interviews for the daily planet, and Jimmy doesn't lift cars. But if Jimmy was interviewing Superman while Superman was saving lives we have a better pairing. It is unrealistic to believe Superman would fight crime with Jimmy. If Jimmy is an innocent bystander however, while Superman is being super, we keep the believably.
Jimmy takes pictures of Superman for the weekly edition, but suddenly a bomb goes off! Superman needs to rush to the city's aid, and Jimmy needs to score a scoop. At the scene of the explosion, armed gunmen attempt to fire on Superman but he is bullet-proof. Jimmy hides behind a brick wall trying to get a closer look. A thug spots Jimmy and aims to kill. Superman flies in and stands in the line of fire! Jimmy yells out thanks, Superman smiles as he rips the assault rifle out of the bad guy's hand, before snapping it in two! Suddenly, Jimmy calls out, Superman look sharp! as a bus-load of school kids hurtles into the fray.
Superman is impossible, Jimmy is not. But so long as Superman sticks to doing the impossible and Jimmy sticks to doing the ordinary, balance remains. I will categorically deny that Superman is a Mary Sue so long as he stays in a “super” story. Let Clark Kent write the editorials.
The last thing I want to mention is about that World's smartest smartest woman from earlier. We can believe there in her, and we can believe that she will fall in love with anybody if its in the right story.
That is where I leave it to you. Tell me, why did the world's smartest woman fall in love with the pastry chef down the street? Providing some context, and essentials for your details is being aware of Mary Sue.