Definitions for "Fanfiction"
(also fanfics): fan-written stories based on animé, video games, e.t.c. The trend probably started with Star Trek, when fans would write alternate storylines or scenarios (usually involving relationships between Kirk and Spock; hence the genre slash (see yaoi), from the way the names of the characters involved were divided), and later, post them online. Fanfics are a way for fans to express their thoughts, opinions or alternate ideas on a show and its characters, or just fulfil fantasies (this latter type of fanfic can be pretty lame: "I love Ryoko! I hate Tenchi! Let's totally disregard the plot and character development and have Ryoko marry some original character who just happens to be a whole lot like me...."). Fanfics aren't strictly legal (being that they use an existing show and its characters), but most authors put disclaimers in their work saying that they aren't making a profit and that the characters don't belong to them, and nobody really minds too much. In Japan, the equivalent of the fanfic is doujinshi.
a story invented and written by a fan and whose story uses existing universe and characters
a written text, so it is normal to have it written correctly