"If I make up my own stories, but base them on another work, my new work belongs to me." False.
U.S. Copyright law is quite explicit that the making of what are called "derivative works" -- works based or derived from another copyrighted work -- is the exclusive province of the owner of the original work. This is true even though the making of these new works is a highly creative process. If you write a story using settings or characters from somebody else's work, you need that author's permission.
Yes, that means almost all "fan fiction" is arguably a copyright violation. If you want to publish a story about Jim Kirk and Mr. Spock, you need Paramount's permission, plain and simple. Now, as it turns out, many, but not all holders of popular copyrights turn a blind eye to "fan fiction" or even subtly encourage it because it helps them. Make no mistake, however, that it is entirely up to them whether to do that.
There is a major exception -- criticism and parody. The fair use provision says that if you want to make fun of something like Star Trek, you don't need their permission to include Mr. Spock. This is not a loophole; you can't just take a non-parody and claim it is one on a technicality.
The way "fair use" works is you get sued for copyright infringement, and you admit you did copy, but that your copying was a fair use. A subjective judgment on, among other things, your goals, is then made.
However, it's also worth noting that a court has never ruled on this issue, because fan fiction cases always get settled quickly when the defendant is a fan of limited means sued by a powerful publishing company. Some argue that completely non-commercial fan fiction might be declared a fair use if courts get to decide.
