Fractals are, as IP states, a concept in Geometry. There are a number of definitions - see http://home.earthlink.net/~mrob/pub/muency/fractaldefinitionof.html but the name, I believe, comes from the idea that a part (or fraction) of the shape when scaled up has the same form as the whole. Alternatively from the idea of an object having number of dimensions which is not an integer, say 2 and a half dimensions not 2 or 3.
Apart from writing little programs to draw Sierpinski and Mandelbrot shapes I know little about the subject. A google will pull up lots of fascinating pictures and articles.