On 14 October 2010, the genius who coined the word - Polish-born mathematician Benoit Mandelbrot - died, aged 85, from cancer.
Unfortunately, there is no definition of fractals that is both simple and accurate. Like so many things in modern science and mathematics, discussions of "fractal geometry" can quickly go over the heads of the non-mathematically-minded. This is a real shame, because there is profound beauty and power in the idea of fractals.
Well, I read his book, and never really understood his explanation. But it is extremely important for earthquakes, and not everybody realizes this. I've also used it to explain why corporations and bureaucracies suck. It's all to do with self-similarity, which is either the defining element of fractals, or a subset, which I can never figure out.
With regard to earthquakes, all the fractal papers are hiding behind the pay wall, and are incomprehensible. I've done my best to explain it, but I hope that others do better. :)