An interesting one, this. Eternally relevant to fanfiction, too.
Okay; to me, a "classic" Sue is a character who is unrealistically good. One who has everything go well with little or no difficulty, who has ridiculous advantages, whose social interactions lack psychology in favour of fantasy fulfillment and so on. There are almost always wishful author avatars. Common traits include extraordinary beauty, being worshipped by those around them, and being absurdly skilled. Often, classic sues will have a single, syuperficial flaw that is capitalised on as 'proof' the character is imperfect.
More insidious sues include those who are self indulgently angsty, who are hated for no real reason by everyone, and yet 'special', or perhaps sues that do fail and do have flaws but the story still revolves around with no concern for the lives or personalities of others, or even the 'justified' sue whose amazing nature is explicable but no less unrealistic or unnecessary.
The mary sue remains one of the most enduring debates in fiction writing because of its nebulous nature and because of the vast spectrum of character design philosophies. The only kind of sue everyone will agree on is the parodic sue that would be considered stupidly and unattractively 'overpowered' even by the most tolerant of readers.
I hold that it is entirely possible to create a character who excels at many things without making them a sue, however! For example: an arrogant prodigy in many disciplines who has no social competence whatsoever and suffers autism, or perhaps schizophrenia. Just off the top of my head, that one.