What is a Mary Sue?

I came on here to clarify, because I think the term has been tossed around too much, and not everyone is gonna understand the reference, particularly if they don't read fanfiction.net or visit TVTropes.com.

See, the absolute very first fanfiction ever written, was a parody story a fan of Star Trek: The Original Series wrote. I'll copy and paste the story here:

The term "Mary Sue" comes from the name of a character created by Paula Smith in 1973 for her parody story "A Trekkie's Tale"[3]:15 published in her fanzine Menagerie #2.[4] The story starred Lieutenant Mary Sue ("the youngest Lieutenant in the fleet — only fifteen and a half years old"), and satirized unrealistic characters in Star Trek fan fiction.[5] The complete story reads:

"Gee, golly, gosh, gloriosky," thought Mary Sue as she stepped on the bridge of the Enterprise. "Here I am, the youngest lieutenant in the fleet - only fifteen and a half years old." Captain Kirk came up to her.

"Oh, Lieutenant, I love you madly. Will you come to bed with me?" "Captain! I am not that kind of girl!" "You're right, and I respect you for it. Here, take over the ship for a minute while I go get some coffee for us." Mr. Spock came onto the bridge. "What are you doing in the command seat, Lieutenant?" "The Captain told me to." "Flawlessly logical. I admire your mind."

Captain Kirk, Mr. Spock, Dr. McCoy and Mr. Scott beamed down with Lt. Mary Sue to Rigel XXXVII. They were attacked by green androids and thrown into prison. In a moment of weakness Lt. Mary Sue revealed to Mr. Spock that she too was half Vulcan. Recovering quickly, she sprung the lock with her hairpin and they all got away back to the ship.

But back on board, Dr. McCoy and Lt. Mary Sue found out that the men who had beamed down were seriously stricken by the jumping cold robbies, Mary Sue less so. While the four officers languished in Sick Bay, Lt. Mary Sue ran the ship, and ran it so well she received the Nobel Peace Prize, the Vulcan Order of Gallantry and the Tralfamadorian Order of Good Guyhood.

However the disease finally got to her and she fell fatally ill. In the Sick Bay as she breathed her last, she was surrounded by Captain Kirk, Mr. Spock, Dr. McCoy, and Mr. Scott, all weeping unashamedly at the loss of her beautiful youth and youthful beauty, intelligence, capability and all around niceness. Even to this day her birthday is a national holiday of the Enterprise.

In 1976, Menagerie's editors referred to the original story, writing:

Mary Sue stories—the adventures of the youngest and smartest ever person to graduate from the academy and ever get a commission at such a tender age. Usually characterized by unprecedented skill in everything from art to zoology, including karate and arm-wrestling. This character can also be found burrowing her way into the good graces/heart/mind of one of the Big Three [Kirk, Spock, and McCoy], if not all three at once. She saves the day by her wit and ability, and, if we are lucky, has the good grace to die at the end, being grieved by the entire ship.[6]