Contrary to what is written there, Mandelbrot did not discover fractal geometry, as he acknowledged. Earlier thinkers such as Lewis Fry Richardson, Pierre Fatou and Gaston Julia have a greater claim (and much earlier thinkers such as Albrecht Dürer might have something to say about it too).
Mandelbrot did however coin the term ‘fractal’ and he helped bring these earlier ideas together into a coherent theory, recognising the potential for powerful applications in many areas of science. He launched these fantastic shapes into popular consciousness through his books the fractal geometry of nature and the (mis)behaviour of markets. His critics might have dismissed fractals as pretty but useless; today, no-one doubts the central role that chaos theory and dynamical systems have in explaining why the universe looks as it does.
[Update: Tom Leinster at the n-Category Café has a very nice technical post about the Mandelbrot set.]
Well, there is only one picture with which to finish off this post: