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 plot-line crashes with the original Naruto series. Sues are usually loathed among OC creators, and are definitely something to avoid while creating a character.
The most common traits that Mary Sues have include: a tragic past, a flawless appearance and an all-out neutral personality. They are usually portrayed to be liked by everyone, as well as being 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 at such a young age, being possessed by demons and 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-colored or two-color 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 to be jealous of her. Other typical traits involve things such as differently colored 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. Most begin this 'gift' with defending themselves from a group of attackers or the Akatsuki, whom most likely murdered their family or something along those lines. They end up with a level higher than Kage or Sage by their adolescence. It is not unusual for them to be a Tailed Beast host, or to be otherwise possessed. Furthermore, Sues often end up dying but are most always revived by some other character.
The relationships are known to affect the Sue the most. Creators of the Sue try to neutralize it by making their character in conflict with a canon character that is automatically unlikable in the series. However, this had been found to backfire; as the reasons for conflicts are often jealousy or revenge. The Sue is usually adored and wanted by everyone, sometimes especially by those who are certainly not looking for love. It is not uncommon for an Akatsuki member to be falling for them. Sue relationships involve OOC (out of character) acting of characters. This could include Sasuke being hyperactive and friendly, Hinata being aggressive and constantly negative, Akatsuki members being caring and gentle, etc. Usually thanks to the Sue's unrealistic and amazing personality/communication skills, Sasuke or another distant character opens up their sorrows to the Sue and most often will fall in love with this 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 humor. 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, even though 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 Hyguua Seal on the forehead,) or pretend that the event hasn't affected them, even though 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.