Some people still use, more than ever, the term to mean any original character, especially an author proxy, but that is an incorrect usage of the term. It is better described as an excessively flawless character who has little to contribute to the story but commands a central role anyway, and it's usually demonstrated by them having to invent reasons why they're not perfect...which are invariably nothing but transparent excuses. Enoby in My Immortal as mentioned by @Aspie_Gamer is a perfect example, although Jenna in the execrable My Inner Life may be an even better one in an even worse story.
A good rule of thumb is that if you're trying to avoid a Mary Sue, avoid characters that are essentially perfect and have nothing to contribute to the narrative. It's fine to have an extremely old or even extremely powerful character -- it's fine to have a character everyone loves -- but there needs to be somewhere the character has to go, to develop, to grow, and something to contribute to the narrative. Perfect characters are not only not that fun to read most of the time, but they're also not that fun to write; when someone has nothing to really do and nowhere to go, they don't add much to the appeal of a chronicle.
There's a difference between a Mary Sue and a character who is just fairly fully-realized; oftentimes such a character will be able to drive the central resolution of a problem while their less capable companions, friends, and assistants experience perhaps more relatable things. A good example of a highly-capable character driving the narrative while not being a Mary Sue type is Bernard Quatermass, for example in Quatermass 2 or Quatermass and the Pit.
One thing to be aware of and to be careful of is the "Batman/Wolverine syndrome", a tendency of writers to use a character as an extension of themselves, to compensate for their own insecurities by overwriting the character. Over the years, both Batman and Wolverine have been used, many many times, as characters that become essentially Mary Sues in a poorly-composed narrative. It's one thing to express oneself through a character, but I think it bears caution when said character becomes a compensation for one's perceived inadequacies.