Context, mentality and dynamic reactions (from other characters).
Context- Make sure the character has a reason to be this way. That's the biggest one, for me. If your context lines up (and it's got a realistic mentality; I'll get to that in a bit) then any sort of improbable skills are believable.
Mentality- If you give your character an amazing set of skills, make sure they have the mentality to go along with it. Somebody who's taken twenty years of martial arts will have a very specific mindset, which should be taken into consideration when writing from the character's PoV.
Dynamic reactions- Don't just have everybody love or be jealous of the character because of their skills. Treat them like every other character when it comes to relationships.
A Mary Sue is a character who is not believable, usually with a skillset that looks unbelievable unless you justify it. Their relationships are usually in two extremes, which is "Everybody loves them" and "everybody is jealous of their skills." They also tend to have no/minimal flaws, or the flaws don't really show up in the story and impact their life.
A writer is a world trapped in a person— Victor Hugo
Ink is blood. Paper is bandages. The wounded press books to their heart to know they're not alone.
There's a second definition of a Mary Sue, and that is "gratuitous author self-insert". These Mary Sues serve the purpose of author wish-fulfillment and are particularly common in fanfiction.
But back on topic:
Think through the consequences of your characters' actions. How do they impact the people directly around the character? How about the far-reaching impacts? Especially in stories involving epic, world-saving plots, the far-reaching consequences need to be taken into account.
Keep track of these consequences. Remember that inconveniences do not just disappear just because they're inconvenient, just as benefits do not stay just because they're beneficial. If a character has been abused throughout their entire childhood, they won't be instantly cured by the first instance of friendship they encounter; the more serious the scenario that impacted them, generally the longer it takes for a character to change their worldview.
Treat all of your characters, including the minor ones that only show up for one scene, as equally important characters. Although you'll never share this information with your readers through the story, it's a good idea to have at least a basic history for all the characters who have significant speaking parts or act in ways that significantly affect your characters.
For instance, if you intend to have a misogynistic bastard in the next inn that degrades your heroine and is a source of her determination to overthrow the oppressive and traditional ruling body of a country, try and portray the misogynistic character with respect. Consider his reasons for acting the way he does, and try to portray him sympathetically, giving his arguments at least one strong reason in which they are based. Not doing so leads to everyone with views conflicting with those of the main character(s) coming across as pathetic, unbelievable, and completely manufactured to make your main character(s) look better.
Make sure your characters' abilities are set and make sense. Nothing makes a reader more likely to throw whatever they're reading across the room than having a character or several with powers that come and go at random, appearing when most convenient and being forgotten when most dramatic. Even worse are the abilities that do the aforementioned and also are parts of skill-sets the characters could not reasonably have, considering their background.
For instance, Generic Peasant A should not be able to read or write, or even possess much fighting ability. However, you have Generic Peasant Heroes all over the place in fantasy that learn how to do all the above and then some in an insanely short period of time. Whenever you have such a break from believability, you break the justification for the character's abilities, which leads to them being seen as a Sue.
Basically, make sure all your characters abilities and actions are justified, even the minor ones. Also, keep track of the consequences of your characters' actions, and have those consequences affect the characters.
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