Essentially there is nothing (I repeat nothing) more irksome in writing than an unoriginal, unlikable, unrealistic character. Before we start, please do not think that your characters musn't carry any of these traits, but as they say, everything should be in moderation. This section applies to Original Characters only, we'll come to Canon Characters in sections 2 and 3.
Name: Something over the top like "Enigma" or "Lo'reyana Darkness". Most likely something that completely stands out from the time, place, culture and society that the character is in.
Appearance: Stunningly beautiful of course. Completely flawless in every way. In fact, other characters only dislike her becuase she is so perfect, but she always manages to bring them round to her side with her kind-heartedness and overall awesomeness!
Personality: Little to none. In a fantasy story, she probably has some kind of special power that no one else has, has ever had or will ever have. She kicks ass, obviously, and probably has a very long list of hobbies and interests, all of which she performs perfectly. Lets not forget that she's clever, even though she spends no time studying and spending all her time oggling over.......
Name: Something dark and mysterious, and full of "meaning". A name that sticks out from the time, setting etc etc. Possibly has a "Z" in it somewhere, because he is just that interesting and original.
Appearance: Long lengthy lists will tell you just how amazingly attractive this guy is! All the girls want him, heck even most of the men want him too! When he walks down the school corridor (Face it, if an OC goes to high school, he is most likely a Gary Stu) all the girls legs seem to just open of their own accord! He also has eyes that are unlike any others, and when he looks at Mary Sue, she can see deep down into his soul, despite the fact that no one looks like that. It's kind of creepy.
Personality: Personalities of the typical Gary Stu never vary far from "the angsty tortured soul hiding under an intense bad boy persona." Or for fantasy stories it's the "tortured vampire/werewolf/demon/fall angel who fights against his blood-thirsty nature, just to be with Mary Sue. Recently been known to drive ridiculously expensive fast cars, even though he doesn't have a job or regular income. He doesn't need to, he's so perfect he is rich by default.
HOW TO AVOID!
Here is an example of an opening paragraph for a fan fiction which is full of Sueness. (Don't forget I have exaggerated this to include most aspects bad characters.)
"My name is Crystal Sky Bellemarie. I have long flowing blonde hair with golden streaks in it. Everyone at my school says that I must have dyed it, but it's completely natural. My skin is pale and I often wear black because I'm so tortured. I'm quite tall, and really slim so I feel really unnatractive. My mother and father were killed in a fatal accident which makes people feel really sorry for me, but instead of being actual friends with anyone, I prefer to sit alone. That was how I was when I saw him...staring at me from across the room. Lorenzo Del Monte....."
Yes, some "writers" genuinely open their stories with this little beauty. Even though I've gone way over the top, there are some writers out there who aren't to far from this. So as soon as you see a story start with "My name is...." then you know to stay far FAR away! (There are some exceptions to this rule. Keep an open mind, I'm just saying, be prepared)