Here is one way to understand this idea. If you try to measure a non-exotic object, like a watering hose randomly tossed onto the garden lawn, with a ruler, then the number you get for the length will depend on the length of the ruler. A six foot ruler will give only a crude approximation to the length of the hose. With a fractal object, like a coastline, the smaller you make your ruler, the longer the coastline appears. This is because smaller and smaller rulers measure smaller and smaller jigs and jags in the coastline.
Fractal objects have jigs and jags on all scales.
They do not start to look smooth as you magnify them. Most fractals are generated by relatively simple equations where the results are fed back into the equations again and again. The recursiveness of this procedure is why one sees structure at one scale in a fractal repeated, perhaps shrunk, rotated, and slightly distorted, on another smaller scale. The procedures that produce fractals can either be probabilistic, like Michael Barnsley's Iterated Function Systems , or determinstic processes that produce chaotic attractors . For some great fractal demos try the Fractal Explorer Kit .