A Mary Sue is a term for an Original Character, often female, with character traits heavily skewed in favor of outstanding attributes that help achieve their goals with little to no effort in proportion to audience expectation. They are almost always the center of attention in their respective fanfictions, and are therefore seen to steal the spotlight from canon characters - original characters who play supporting roles in fanfiction are very rarely true Sues.
This character type is mostly associated with fanfiction, though it can be found in original fiction as well. Female characters of this type are the most common in South Park fanfiction due to female fanfic authors being more numerous than male, and a tendency for younger authors to focus on their own idealized traits. Within the fandom, they often form romantic relationships with various Canon characters.
The term is frequently misused as a misogynistic slur to any original character, regardless of personality, usage, gender, or other factors, and the woman who coined it has since come out against the term as a result.
The main association with a Sue is that the character will achieve their goals with little to no effort and experience minimal obstacles that are often trivialized by the narrative. In most cases, Sues will be instantly liked by favorable Canon characters, such as Eric Cartman, Stan Marsh, Kyle Broflovski or Kenny McCormick, regardless of circumstance, and often causing them to act out of character. If a Sue suffers from an antagonist, they often dislike them for inexplicable reasons or see them as a rival for another character's affections. Sues do not undergo character development and are static characters from the story's beginning to end. It is common for Sues to be 'flawed' by being clumsy or blaming themselves for events in their past that were not their fault, such as a divorce or abuse, which are often either serious but trivialized, or non-serious to begin with. Physically, they may have a lone birthmark in a peculiar shape or small unique decoration.
Some Mary Sues are often characterized by having supernatural, inhuman talents that carry little narrative weight, providing primarily benefits and causing only brief angst with little long-term bearing on the story, often as a way to give the character a brief flaw and/or to make them more "special". While characters with strange abilities are not unheard of in South Park, such as Sir Douchebag in South Park: The Stick of Truth, these abilities must carry thematic and narrative weight. Mary Sues can also possess extreme beauty, muscles, strength, agility or commonly or singing talent. Some of them may even best canon characters in expertise, such as overtaking Kyle or Wendy for smartest kid in school, often a sure way to annoy readers by further hogging the spotlight.
While a relationship between a canon character and an original character does not inherently make a Sue, these relationships are nonetheless frowned upon in South Park fanfiction and generally will draw accusations, even to otherwise good characters. In Sue stories, the romantic partner often becomes a satellite love interest who exists mostly to please and support the Sue character, with little to no will of their own, and almost always providing a voice of comfort, rarely arguing or debating with them on any topic. The only obstacles to their relationship will often be outside factors; a Sue focused on Stan Marsh may, for example, be targeted by a jealous Wendy Testaburger, or blocked by an angry Randy Marsh. Sues are often accompanied by florid over-description.
Some original characters are Non-canon relatives of existing canonical characters. While this is not an inherent trait of a Sue, and can be done well in fanfiction, it has nonetheless become an increasingly common practice among Mary Sue writers as well, as it can be an effective plot device to explain why the new character is accepted by the main characters without question, rather than explaining how the new character wins the existing characters' approval.
Please note that Sir Douchebag from South Park: The Stick of Truth, for example, inherently possesses many masculine Sue-like traits, including a special ability, minimal angst, variably strong obstacles, and being the center of attention. While some of these traits are almost certain to appear in a given original character, and none alone inherently make a Mary Sue, characters who possess all or even most of these traits mentioned in this section should be evaluated with caution.
Male characters of the same type may be called Gary Stu, Marty Stu, or Marty Sam; all of these are synonymous plays on the feminine blanket term. "Stu" is the common short form.
"Airy Ooh," another play on Mary Sue, is a proposed gender-neutral term for the same character type, but it has almost never been used.
Online Mary Sue Tests
There are many sources online to help writers identify when they've created a Mary Sue. Taking a simple "Mary Sue Test" can be useful in identifying if your OC is a Sue and what characteristics to watch out for in the future.
Here are a few links to some online tests. Keep in mind that these tests are not definitive sources for identifying Mary Sues, although they're still a good place to start.