An Original Character (OC, don't steal) is, in the simplest terms, a new character created in a Fanfic or other work that does not come from an existing (copyrighted) property. Any and all Characterization Tropes can apply, along with employing any and all tropes in general. The only real distinction between original and regulars character is that the former are synthesized specifically to unofficially integrate with the canon for the purposes of the story. The vast majority of fanfiction makes use of these, ranging in importance from being background extras to stealing the spotlight of the canonical characters.
On that note, original characters have a bit of controversy surrounding them. Most people read fanfiction to see the canon characters and feel cheated when they find that the story basically reduces them to satellites in favor of what is often a Mary Sue, especially when there are already enough characters as it is. This isn't always the case, of course, but Sturgeon's Law means the odds are good.
Still, Tropes Are Not Bad and perfectly valid, likable new characters can come about. After all, all characters were new once. Series that are structured around new characters every week such as Monster of the Week, Victim of the Week, or Girl of the Week actually require that the author create original characters in order to maintain the Original Flavor of the series.
Attempting to list every instance (or even every "notable" instance) is rather pointless—they are almost as numerous as Fanfic themselves.
In some places (such as on Deviant ART) the term "Fan Character" is used instead, and the distinction "Original Character" refers to a character that exists in a canon of the author's own creation, or the canon of an "Original Character Tournament" (where artists compete by pitting their original character against other peoples' in fights, or pizza-eating contests, or whatever the creator of the tournament has decided is the proper form of conflict); in other words, an "Original Character" is a truly original character without ties to another creative work's canon.
One general mental comment to the "OC, Do Not Steal" disclaimer is the reaction, "Why do you think I'd want to steal your character?"