Mary Sue... Ah. I've seen those a lot. Well, let's begin with the analysis, shall we?
The term has been thrown around so much it's hard to pin exactly what the term entails, but I personally consider it to be these traits:
1: The character is infallible, or almost always right.
Now, there's no problem with character having ideas that mesh with reality, or predicting future events, but when every single plan goes perfectly, and every single prediction comes true, then that's a problem. Characters are interesting when they fail, too. In fact, that brings me to my second point.
2: Never seriously getting harmed, refuted, or refused.
Again, there's nothing wrong with your characters NOT being barely alive piles of wounds and regret, or getting what they want every now and then (Something I myself needed to learn the hard way), but, if your character has no regrets or suffers no injuries (or always recovers perfectly), then you're entering territory where the reader says: "okay, this is a Sue,", and you don't want that. Further warning signs include being everyone's friend, and getting the most antisocial characters to trust and adore them.
3: Unrealistic appearance, intelligence, physical ability, wealth, with no drawbacks or flaws.
This is the trademark, shut the door, final nail in the coffin for most characters. Abilities or attributes that are unrealistic. Here's an example, with two character bios. One of them is a Sue. It will be obvious which one is.
"John Seback is a thirty-two year old man. He is rather thin and tall, and his wiry brown hair seems to have a life of it's own at times. He has brown eyes and a rather large nose, which, when he was younger, was mocked often. While not particularly athletic, he occasionally plays basketball, and is rather skilled at it. He doesn't often talk about his family, or talk much to people he doesn't know. He still finds his nose embarrassingly large at times."
"Maryssa Lynafell Raven is an eighteen year old girl. She is exactly 6 feet tall, and has long, straight, sparkling violet hair that glows like the sunset. Her body is perfect, and she is skilled in rock climbing, sword fighting, chess, computer hacking, piloting any sort of aircraft, and is stronger than most adult men. She also enjoys playing with her half-dragon pet, and competing in fashion contests, which she always wins, of course, she also has..."
Actually, that mockup brings me to...
4: Overtly long or poetic names.
If you had a kid, you wouldn't name them "Ebony Raven Dementia Tara Way", you'd probably name them "Franklin Joeseph (whatever your last name is)"
Long, unique names (if everyone has a Spanish name, a character with a Spanish name isn't even on the radar, so to speak) are often a sign of being a Sue.
5: Overtly long or poetic descriptions.
Let's say you have a friend named Jim, and someone asked you to describe him. Would you say:
"Jim's about average in height, and is somewhat heavy. He has brown hair and large glasses, that somewhat make him look like an owl. Everything he wears is brown."
"Jim is exactly 5'7, and is EXTREMELY THIN AND FIT. He has gorgeous blah blah blah blah blah..."
Describe your characters as if you were describing them to a person. Do not describe them to yourself.
And those are the basic warning signs. I find it hard to read about characters without flaws, because I just feel like I am supervising a little kid's inner fantasies, with the character representing the child. No flaws. No worries. Just them and their whims. And that's not fun to read.