Fanfiction is a work of fiction created by a fan of the original property without authorization from the original creator or current owner of the property.
Fanfiction has existed at least since the early 17th century, when sequels to works such as Don Quixote were written and published by creators other than the author of the original. Depending on your definition of "fanfiction," the practice may date from as early as 800 AD, when various authors published their own takes on the tales of King Arthur. The modern era of fanfiction began with the Star Trek fandom in the 1960s, which saw the beginnings of magazines created by fans for fans containing fiction inspired by the TV series.
The internet has helped fanfiction to grow in popularity. Between dedicated fanfiction websites, blogs, and free webspace, virtually anyone with internet access can post their writing for other fans to read. Websites like this one, with detailed information on popular TV shows, books, films, or comics, facilitate the research needed to write a story that is true to the original property. Because the internet allows fanfiction authors to provide their stories to readers at no charge, it also allows many authors to fly under the radar of creators or companies that would take legal action if fanfiction authors were making money from their creations.
Why Write Fanfiction?
The vast majority of fanfiction is never published in the traditional sense. The only exceptions are licensed novels, which may not even be considered fanfiction depending on your definition. For legal reasons, fanfiction authors cannot make money from their works. While some creators take a very positive view of fanfiction based on their works, there are others who actively discourage it. So why put your effort into a story that you can never profit from financially?
For many authors, fanfiction is a labor of love. These writers love the original property and want to express their affection for the characters and concepts by writing their own stories about them. When a series ends prematurely with unresolved storylines, as Gargoyles did before the comic book came into being, fans will often step in and write their versions of what happened next. Some fans write to deal with something they don't like about an otherwise beloved series - the death of a character or the romantic pairing of two characters - by writing an alternate version of events more to their liking. Fans may choose to flesh out the story of a character who never got much play in the original fiction. Some writers create original characters to join the canon cast. These characters are sometimes stand-ins for the authors themselves, allowing fans to interact with their favorite characters. Both novice and experienced writers can use fanfiction to hone their writing skills or simply to play with a world that has already been created. Though original creations will and should always be the goal of most professional writers, the ability to work within an already created concept can be a very useful skill for writers working in modern media.
With Gargoyles in particular, many fans wrote out of a combination of inspiration and frustration. Though many writers started crafting their own stories of the Manhattan Clan and their friends and foes during the show's first and second seasons, a number of fans became fanfiction authors due in part to frustration with The Goliath Chronicles, a third season made without the participation of Greg Weisman and now considered apocryphal. The ending of the TV series, combined with the revelation of Weisman's Master Plan for continuing the series and creating spin-off shows, inspired many fans to fill the void and pen their own versions of what happened next.
Greg Weisman and Gargoyles Fanfiction
The attitude of creators to fanfiction based on their works varies from individual to individual. Some creators give their fans free reign to write stories set in the worlds they have made. Some set out strict rules for their fans to follow. Others take no stand about it either way. Still other actively discourage it and a few even actively work to stamp it out.
Greg Weisman has somewhat mixed feelings about fanfiction.  On the one hand, he is flattered that Gargoyles has inspired so many people to write stories based in the Gargoyles Universe.  He also realizes that fan-created work has helped to keep Gargoyles alive between the end of the TV series and the beginning of the Slave Labor Graphics comic.  On the other hand, Mr. Weisman feels somewhat protective of the Gargoyles Universe and encourages all aspiring writers to create their own original works of fiction. 
Greg Weisman has a long-standing strict policy of never looking at any fan-created work.  Like many creators, he does this for his own legal protection.  If a creator reads a piece of fanfiction or has an idea suggested to him or her by a fan that bears some similarity to something the creator was going to do in the canon fiction later on, the fanfiction author could conceivably sue the creator for "stealing" the idea.  The only way for a creator to completely avoid this potential legal entanglement is to avoid all fan-created work based on the property. Mr. Weisman has been sticking with this policy for years, and it is not negotiable.  He will not read your fanfiction, listen to your ideas for a fanfiction, help you write your fanfiction, or have anything whatsoever to do with your fanfiction. Not ever.