If you want something done right, do it yourself.
Fan Fiction, is a form of Alternate or Expanded Universe created by the fans of a work, rather than the works original creator. Fan Fiction, or "Fan Fics" as they are often called, are written for several reasons: To continue a story that ended prematurely, to see what would happen if certain characters are placed in unusual situations, to see what happens when the characters of one franchise encounter the characters of another franchise. Or sometimes to get get the chararacters to have mad passionate (or occasionly just mad) sex with each other.
Due to the inexperience of many Fan Fic writers, Fan Fiction has gained a reputation for being a source of horrible, Horrible writing. However there are Fan Fics out there that are INCREDIBLY good. Often being just as good, if not better, than the original work.
Saying "It was a Fanfic episode," though, is not usually a compliment.
Some franchises -- such as Star Trek -- have actually turned fan fiction into a profit center by creating Tie-in Novels. These books are usually penned by young and upcoming authors, often former Fanfic writers, and represent an intermediate step between fan fiction and completely original fiction.
Although Fanfic exploded along with the Internet, it existed well before the Net did. Such luminaries as John Stuart Mill contributed unauthorized, original stories set in a fictional universe. Before medieval French troubadours were shipping Launcelot and Guinevere, the ancient Greeks were writing plays about relationships between characters in The Iliad. In Plato's Symposium one character complains that a play by Aeschylus got the characterization of Achilles and Patroclus wrong -- namely, that it got the Lover and Beloved dynamic backwards.
Not all Fanfic is written, though that's the most common form. It can be in any format that can tell a story. In Japan, Doujinshi (amateur "comic books") is a common vehicle; and with the increasing ease of their production on personal computers, fan videos (ranging from Anime series, to Star Wars) have already appeared.
The distinction between Fanfic and original fiction, as we know it today, is largely created by modern copyright law; much of classical writing is actually "fanfiction" based on older sources. The major distinction between fanfic and a story inspired by another story is that the story a fanfic is based on has one or more "official" versions, usually owned by a company, a creator, or both. Thus, things like The Infancy Gospel of Thomas, a piece of biblical apocrypha featuring Angry!Uber!Baby Jesus, or variations on Arthurian Legend where there is no Holy Grail and Lancelot's affair with Guinevere never happens, would not "count" by this definition.
No statement on the legality of Fanfic has ever been given in American formal law or in its courts. Some argue that it's a form of copyright infringement; however, see "Legal Fictions: Copyright, Fan Fiction, and a New Common Law", and note the above precedents.
Authors often have conflicted reactions to fan fiction set in "their" universe, which sometimes leads to a Fanwork Ban. J. K. Rowling has largely embraced Harry Potter Fanfic, albeit with certain limitations, for example. By contrast, George R. R. Martin, author of the epic fantasy series A Song of Ice and Fire, expressed his disdain for the practice, saying that "creating your own characters is a part of writing." He's even gone so far as to threaten legal action should he become aware of any fan fiction set in the Westeros universe. In contrast, writer/journalist James Bow makes a rather firm case for supporting Fanfic, pointing out that it forms a stepping stone towards creating your own characters and setting.
Authors do have non-emotional reasons to worry: as Mercedes Lackey had long taken pains to point out on her website, a fan writer was once able to wrest control (via a successful lawsuit) of part of Darkover away from its creator, Marion Zimmer Bradley. This is the ultimate nightmare of any writer, fan or professional, and drives some of the more draconian efforts to suppress fan creativity. (Lackey herself was once infamously on the draconian side of the divide in part because of this event, but in early 2010 reversed her stand on fanfiction thanks to an association with Cory Doctorow.)
In the end, more and more media outlets are recognizing that fanfiction and other fan works are a simple fact of life. And as art imitates life, it's now possible to find "fake" fanfiction created as part of a marketing campaign. For example, the 2010 season of Showtime's The United States of Tara introduced a new character, an artist who had created and published a comic book character named "Princess Valhalla Hawkwind". As part of the promotional buildup for this, Showtime actually created a "fan site" complete with fanfic, fan art, and fan video.
Some fanfiction becomes well-known enough to influence other fanfics, which themselves influence more fanfics, and so on in a domino effect. This can and does result in the creation and perpetuation of Fanon, when one author comes up with a "cool detail" and others blindly copy it without realizing it was his invention. Furthermore, characters can become Flanderized by the feedback loops of fanfiction, sometimes changing dramatically from their original form.
Eventually, this accretion of fan-born details and mutations turns into things that "everybody knows" about the series. Those new to or unfamiliar with the original material are frequently confused into believing that it obviously must be Canon if so many people mention it, even "facts" of the Epileptic Trees variety. This is especially the case with series that have long runs and which gloss over details which are unimportant to the plot but are of interest to the fans and the fan writers.
One famous example of this is the anime Ranma ½, released well before the Internet became ubiquitous and when many fans had no easy access to the original source material. All manner of details (including the explanation of Akane's mallet as either a ki attack or as residing in a hyperdimensional pocket, her Flanderization into a "psychobitch", her lethal cooking (rather than being just bad), and the names and fates of the many missing mothers) were never touched on in the show but became standardized in Ranma fan fiction over the course of approximately a decade. The process was accelerated and exacerbated by the appearance of fanfiction written by people who had never actually seen the show itself and whose only exposure to Ranma was other fanfiction.
Another famous example is the Harry Potter fanfic The Draco Trilogy, which was apparently so widely read that details such as Blaise Zabini being female and Ginny's name being Virginia were taken to be canon, although they were both refuted by later books.
It's not surprising that fans of some shows occasionally pen FAQs solely to reduce the accumulation of Fanon in this way.
Currently, the largest source of fanfiction on the Net (and probably anywhere else) is the aptly named Fanfiction.net, which offers a couple million stories across all but a select few canons (which were banned due to creator request) and an automated system for posting. While other sites exist, no other site offers as large an audience.
For fanfic-specific tropes see Fanfic Tropes. Of course, the hive mind have a few favorites. And a few unfavourites. Not to forget a few favorite unfavorites, if that doesn't confuse you too much. Some here have even written a few. And here are some that people took the time to make a page for.
- After Action Report
- Log Fic
- MST (verb form of Mystery Science Theater)
- Poem Fic
- Script Fic
- Song Fic
- Accusation Fic
- Crack Fic
- Curtain Fic
- Dark Fic
- Fusion Fic
- Gen Fic
- Hurt Comfort Fic
- Kid Fic
- Next Gen Fic
- Period Fic
- Porn Without Plot
- Recursive Fic
- Slash Fic
- WAFF (Warm And Fuzzy Feeling)
- War Fic
By relationship with canon:
- Alternate History (where "canon" here is Real Life)
- Alternate Universe Fic
- Deconstruction Fic
- Elsewhere Fic
- Fix Fic
- Hate Fic
- High School AU
- Original Flavour
- Patchwork Fic
- Peggy Sue
- Revenge Fic
- Troll Fic