A Fan Character (FC), also known as an Original Character (OC), is as the name suggests a character that has been created by the fan of a certain series, for inclusion into that fan's fanfiction or fanart. Fan Characters can also be shared among two or more fans, or between a fan and an illustrator.
Fan Characters are often disliked for their association with the "Mary Sue/Gary Stu" subset of character, but this is more the hallmark of bad writing than any negative reflection on Fan Characters as a whole. Some people, naturally, would disagree with this sentiment.
Widely Known Fan Characters