What is a fractal? Here's a definition:
Mathematician Benoit Mandelbrot's term for shapes that are "self-similar," appearing the same at different magnifications. A fractal can be created by duplicating a shape successively according to a set of rules. The results can be complex structures, which resemble seemly random-shaped things in nature, such as clouds, trees, and mountains. An application of fractals is to represent complex imagery in very concise algorithms.
Does that help at all? I didn't think so. Basically, take a mathematical formula, have your computer crank through the different X and Y variables, and you get a cool picture. Then zoom in on a section of the image, and you get the cool picture again! And with even more detail. Then zoom in again. And again.
These fractals were generated with Fracture, the first shareware piece of software I ever personally registered.