Mary Sue or Gary Stu / Marty Stu are character archetypes which symbolize wish fulfillment on the part of the player. TV Tropes has an amazing article on the history of the term and where it came from. The character archetype is generally thought to be female (especially by its name) but it is important to note that this isn't actually so in actual role play.
From here on out when we refer to Mary Sue we are not being gender specific.
Cannot Suspend Disbelief
A Mary Sue character is one that is an obvious mirror to everything amazing in the players mind. Usually going entirely over the top in all areas. This is not to be confused with powerful characters. A Mary Sue is just too good to be true, amazing is every way with little or no flaws. A better way to put it is that the character is very powerful but rather two-dimensional with a lack of explanatory backstory or flaws to balance out the power.
Bombshell / Stud
A Mary Sue type is almost always a bombshell or complete stud embodying everything that the player believes is most beautiful and desirable in looks. Skinny, long legs, petite, perhaps huge breasts, red hair, not an ounce of fat in the female archetype. In the male, a 6-pack, muscular (but doesn't look like a body builder), perfect smile, blue eyes, strong jaw, etc... Both genders mix this with incredible amounts of self confidence even if it makes no sense given some tragic backstory.
Always Virtuous
Everyone has flaws. Even the immensely powerful Superman has the flaw that he is a boy-scout type of personality who tends to see the world in black and white, good and evil, with little room for shades of gray. Such flaws help to balance out a high powered character and limit the actions they would undertake. Similarly, Batman is not so virtuous but far less powerful. Tony Stark is limited by his public face and public knowledge of his person leaving society to reciprocate should it venture too far outside his role - he is also an egotist and narcissist with problems connecting to others personally.
A Mary Sue type of character lacks these images of depth. Spiderman is famous for the line, "With great power comes great responsibility." this is similar in some ways as with great power comes great problems. Heroes are born out of misery, tragedy, hate, anger, abuse, and the like. It is like expecting a rock star to write a song that will live forever when the life of the rock star was perfect in every way. Without these traits the character comes off flat without depth.
Without flaws to offset power then the character is impossibly always virtuous in every way.
Too Many Roles
I was once talking with someone about an RP they were running where a new character applied with the race of (and this was not a joke); Ninja, Vampire, Seer, Werewolf. Often a Mary Sue type has too many roles that they are experts at. You can have a renaissance type of character with a huge breadth of knowledge, but isn't an expert in everything. Often times you'll note that a Mary Sue type is some kind of amazing fighter, brilliant scientist, is thousands of years old, has incredible amounts of money, is famous enough to get into doors that are closed for everyone else but not so famous that they have the down-side perfectly balanced.
Been Around Too Long
To be an expert at something, someone has to put the time into it and be trained, have practice, etc.. It is very rare for even a long-lived race to have so many focuses in life beyond what they choose to do. If someone is 1000 years old, would they major in collect in everything, or pick one thing and make a career, then pick another thing and make a career, etc... This is where the immense old age comes in, a character whose life spans eons and eons would have downsides and issues that you don't see in a Mary Sue.
Doesn't Fit
Overall the biggest indicator is that the pieces don't fit together. Amazingly beautiful, incredible intelligence, inconceivable skills in an enormous array of talents / superpowers, a life entirely too long, and all wrapped up into a package with a serious lack of flaws. These all point to a Mary Sue / Gary Stu type.
Converting Away from Mary Sue
Players sometimes realize at some point that characters are very much a Mary Sue. It isn't necessary to throw these characters away. They can instead be converted to something more palatable. Usually this means weakening the characters, adding flaws that really match the consequences of the character's abilities. Create a layered backstory that gives depth to the personality. Ask yourself what a character believes in, what they love, what they hate, and build from that to give depth to the personality. Then expand on the backstory and add flaws for each ability, then choose a core set of abilities to be very good or amazing in and focus on those.