Robert P. Munafo, 2008 Feb 20.
(This history is partial; additions are welcome.)
In 1918, Fatou and Julia, while working towards a prize to be awarded for "study of iteration from a global point of view", studied the iteration of f(c) = c2 + K for complex c with K constant. They each independently discovered many of the properties of Julia sets and Fatou dusts.
Julia did not produce any images of the Julia sets himself; the first such image appeared in a later paper by Cremer.
In 1978, Brooks and Matelski, published the first image of the Mandelbrot set's interior. They did not investigate it to anywhere near the extent that Mandelbrot would a few years later, and did not know much about its properties. In particular, they omit the filaments and (probably) the islands.
A year or two later, mathematician Benoit Mandelbrot discovered the Mandelbrot set during his research of the Julia and Fatou sets. His discovery was independent of the work by Brooks and Matelski. Mandelbrot describes the Julia sets and the Mandelbrot Set (which he calls the "mu map") in his book The Fractal Geometry of Nature. He also gives a brief description of its original discovery.
The discovery made its way throughout the mathematics community and exploration of the Mandelbrot Set was eventually taken on by Peitgen and Richter, ARTMATRIX, and others. The images of Peitgen and Richter became known to A. K. Dewdney who then used them as the subject for his column in Scientific American, which appeared in the August 1985 issue, featuring one of the images on the cover.
The article was seen by many people and immediately created a widespread interest in Mandelbrot Set computer programs.
From the late 1980's onward there have been dozens of computer programs to draw Mandelbrot and other fractal images. FRACT386 (soon renamed "FRACTINT") was particularly notable for its arbitrary-precision capability and use of integer math to speed calculation on the non-FPU equipped computers of the day. My own Super MANDELZOOM was faster still, but only available for Macintosh.
Efforts to determine the area of the Mandelbrot Set began in the early 1990's.
The late 1990s saw the debut of XaoS, notable for its ability to zoom in real-time at frame rates rivaling a videogame.
Lecture notes from Harvard mathematics course 118r "Dynamical Systems" (Oliver Knill), Spring 2005.
WikiVisual, Mandelbrot set.
revisions: 20080220 oldest on record; 20100921 add software references
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2012. Mu-ency index