Nobody enjoys the sinking feeling that accompanies the end of an amazing book series or the final episode of a fantastic TV show's season. Where can you turn with your now finely-tuned addiction? With the internet, you never have to worry about a great book/Movie/TV show ending. Welcome to the world of fan fiction.
A fan fiction is a recreationally-written story inspired by a nonfiction work or real-world celebrity. For example, a Harry Potter fan fiction may take place after/before/in-between the Harry Potter books, within a different magic school never covered in the books, from the perspective of random muggles living their daily lives with the events of the book happening around them, in an alternate reality in which Harry and Hermione get married, or in an alternate reality in which the Dursleys were fanatical atheists.
Despite stereotypes to the contrary, fan fictions can be quite entertaining and of very high quality. It is not uncommon for fan fictions to be better than their source material. Continuity and fluctuations in tone are less of an issue with fan fiction than you might imagine. Consider that your favorite TV Shows are written by a large, revolving group of writers. The multiwriter nature of TV Shows makes many works of fan fiction seem perfectly at home within the rest of a series. In many cases, the only difference between a fan fiction and a canonical manifestation of a fictional universe lies in its creators' willingness to pay for official rights to the brand (this is often the case with Star Trek [obviously, a gross generalisation]).
Some fan fictions even pick up large fandoms of their own, such as the Fallout Equestria cross fic (a cross fic is a fan fiction inspired by two or more universes). This fan fiction, combining the cute and cheerfully universe from My Little Pony: Friendship is Magic with the bleak and violent universe of the Fallout video game series, has inspired hundreds of derivative fan factions (one over a 1.4 million words).
Though many view the concept of fan fiction to be relatively new (i.e. something that has emerged within the past century), most of our history's literature could be described as "fan fiction" (the entire Greek pantheon reads like a bunch of fan fictions). Writing about popular folk heroes, mythological figures, or other popular fictional characters was so common it was not seen as a different type of writing. The concept of strong character ownership is a fairly recent concept, and has, to a great extent, forced us to see fan fiction as a separate genre. The growing prevalence of fan fiction websites and availability of fan fictions in the form of ebooks has only made the genre more prevalent and visible.
The largest fan fiction site is fanfiction.net. Unfortunately, the site is difficult to navigate and not a great place to find your first fan fiction reads. Some fandoms have their own fan fiction sites which are much better made, easier to navigate, and have better rating systems (ie. fimfiction.net), but given how diverse the number of fandoms out there, there is little need to limit your scope by pointing in any single direction. As simple as it may sound, the best way to find good fan fictions is to just Google, "best [insert your favorite book/TV/movie here] fan fictions".