Regular text is linear - you begin reading at one word and then proceed to the next. From any given word, there is only one other word that you can go to (if you want to understand the meaning of the text). As a reader, you are trapped in the same way that the inhabitants of Lineland are trapped in their world, with only one degree of freedom (which amounts to very little freedom).
Hypertext is text with an added dimension. Words in one passage are linked to another passage, in the way that a link in a Web document connects to another document. This means that from a linked word, you have two choices of movement instead of one - you can continue on to the next word on the screen, or you can follow the link to another screen. The dimensional analogy may be somewhat weak, but I think that hypertext is an accurate description of this genre.
I have composed a small hypertext story to give you an idea of how it works. Writing in HTML (HyperText Markup Language, incidentally) is rather unwieldy, so the links are fairly limited. Fortunately in my class we use a software program called Storyspace which is much more writer-friendly than HTML.
For some more quality hyperfiction, visit Hyperizons or Tree Fiction on the World Wide Web.