The Julia Set and Mandelbrot Set are sets on the complex plane that create pretty infinitely detailed images. The Mandelbrot Set isn't a real fractals by definition, but it's semi self similar and still shows infinite detail, so it's usually called a fractal as well.
Research on Julia Sets was done in 1917 by Gaston Julia himself, but he didn't have a computer available to actually draw it. The topic didn't gain much interest, until in the 1970s, Benout Mandelbrot drew Julia Sets on a computer, and discovered the Mandelbrot set.
They're so pretty, that there is even art created with them. (Taken from: http://lodev.org/cgtutor/juliamandelbrot.html )