The idea of hyperfiction is to take advantage of the computer's flexible text access and presentation capabilities to allow the reader a more active role in the process of interpretation. Typically text is presented a small amount at a time (usually one screen-full or so), and it is possible to click on or type in words or phrases that determine which passage is presented next. The reader thus skips around through a large collection of segments as determined by the choices he makes.
Even with conventional text it is clear that the reader's interpretation is as much the result of idiosyncratic factors in the meaning construction process as of universal ones. Hypertext allows these idiosyncratic factors more play by letting them determine, to an extent, the subset of the work that is actually read and the order it is read in.
This presents new challenges to both the author and the reader. For his part, the reader must be prepared to integrate a nonlinear, possibly disjoint sequence of text fragments into a coherent structure. The author, for his, must arrange things so that this is feasible along all of the different paths that may arise, and he must be willing to accept that he has less control over the space of the reader's possible interpretations than is the case in more traditional forms. This calls for a different strategy in the conveyance of meaning.
There is much more to be said on hyperfiction generally, and some of it has been said and written up in various places (see below). Perhaps the main thing to realize is that it can take many forms, depending on how the author balances and uses the interplay between text and link. Some instances are more "literary", others more "recreational", and the most interesting blur this distinction considerably. Text adventures such as the Zork series are examples of recreational hyperfiction, and a collection of more literarily-intentioned works are available through the Hyperizons page referenced below.
Check out A Further Xanadu, my own attempt at a work of hyperfiction.
See the Collaborative Fiction and Monoblogging page.
Go to the "On-line Books Page", which is a reference-of-references to lots of hypertext and online non-hypertext. There's enough stuff accessible through here to keep anyone occupied for quite a number of years.
Finally, if you like adventure games, here's a good page to check out.