I know that one example of Mary Sue I heard recently was the lead character of the Twilight novels. Someone had been doing a multi-part analysis of the Twilight books, and a large chunk was explaining why the main character was a Mary Sue.
She looks a lot like the author in terms of how she's described. Everyone either loves her or hates her ... and in most cases for no given reason. The character is good at everything, except for a minor flaw (in this case clumsiness) that rarely ever is actually bad for the character, and in fact, is an endeering quality for her love interest (the one she likes back, as opposed to every other guy who is also in love with her). Much time is taken to have people talk about how pretty/unique she is while she denies it.
One of the biggest issues is that a Mary Sue is incapable of having a meaningful character arc, because they are already so good/awesome/flawless that they can't really overcome obstacles and grow as a character.
There are different problems between Fan Fiction [or any adding to established fiction ... Chloe Sullivan on Smallville, for example, was a potential Mary Sue as she was a 'new character' added to the Superman mythos and at points in time was the girl that would be 'perfect' for Clark. Her character, however, has had sufficient flaws brought up, etc] and a new character in it's own work. RPGs have another issue as the problem with Mary Sues there are less to do with litterary issue as there is a risk of either a DM PC upstagging the players, or one player's PC being a spot light hog (or "DM's girlfriend/son/etc's PC" who has extra layers of plot immunity or uber-important artifact, etc).
In a tabletop game, certain things that would be ok in a normal story are more greivous, but in other aspects, the game table is more lenient. It's hard to avoid having PC's have tinges of Mary Sue, as long as the DM doesn't let them get too overboard. Any NPC that overshadows the group is going to build resentment, especially if it is on your side (an enemy that's better than you is a challenge to eventually beat).