A Mary-Sue (f) or a Gary Stu (m), is a 'perfect' character. In OC's it usually involves unrelated plots, God-like strength and appearance, OOC (out of character) acting of Canons and plotline crashes with the original Naruto series. Sues are usually loathed amongs OC creators, and are something to deffinitely avoid while creating a character.
The most common traits that Mary Sues have include tragic past, flawless appearance, all-out neutral personality. They are usually portrayed to be liked by everyone, as well as overly powerful. The Sues tend to be 'special', in the terms of getting a lot of attention because of their skills, for example being asked to become a Hokage, being dragged to the Akatsuki or such at a young age, being possessed by demons etc. The traits can usually be neutralized, making the Sue a balanced character.
Sues are often seen with physical traits that are quite unusual for Naruto Universe, such as cat ears, tails, neon-coloured or two-colour hairstyles, gothic fashion and such. Their appearance is usually very attractive with the Sue in most cases being oblivious to the fact. Other characters tend to be flaunted by the Sue, or be jealous of her. Other typical traits involve things such as differently coloured or demonic eyes, clothing that is uncomfortable for fighting (if the Sue is a shinobi) such as long kimonos or an overload of accessories, or an extreme lean to a particular subculture such as the gothic, emo or 'scene' fashion.
The Sue is usually a very gifted fighter with extreme strength and endurance. They usually begin this 'gift' with defending themselves from a group of attackers or the Akatsuki, who murdered their family. They end up with a level higher than Kage or Sage by their adolescence. It is not unsual for them to have be a Tailed Beast host, or to be otherwisely possessed. Furthermore, Sues often end up dying but being retrieved by some other character.
The relationships are usually the most important part, with the Sue being involved with everyone. Usually creators try to neutralize it by making them in conflict with a character that is automatically dislikable in the series, however this usually doesn't work very well as the reasons for conflicts are often given as jelousy or revenge. The Sue is adored and wanted by everyone. It is not uncommon for an Akatsuki member to be falling for them, thus Sue relationships majority of time involve OCC (out of character) acting of characters, such as Sasuke being hyperactive and friendly, Hinata or Naruto being agressive and with a negative attitude, Akatsuki running around in their underwear etc. Usually thanks to the amazing personality and communication skills, Sasuke or another distant character opens up their sorrows to the Sue.
Sue is usually a depressed person because of the horrible and drastic past. However they are still hyper and have a great sense of humour. They are understanding and comforting towards everyone, and as mentioned above, often have characters like Gaara or Itachi pour their souls out to the Sue. She is caring, kind and sadistic. A double personality is not uncommon either, with the demon taking control of them and making them an evil masochist/sadist.
The Sue plots usually do not involve anything but relationships and battles. The Sue, eventhough being an Akatsuki member, does not chase after any Tailed Beasts. Other tragic events such as deaths, gory scenes of torture or the drama of confessions to their beloved are not uncommon. Often the writers seem to be too lazy to do their research and make their OC's past involve events that happened years before they were born, lacking clan traits (for example the Hygua Seal on the forehead) or pretend that the event hasn't affected them, eventhough it definitely would. E.g. The Sue is 16 in Shippuden, and their parents were killed during the Second Shinobi War, which would make no sense considering the OC would be much older than 16 if they were born around Second Shinobi War. Narutopedia is a great source for plot information.
A fake example: Aiko Kirei