This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.
The Mandelbrot set iterates z^2 + c with z starting at 0 and
varying c. The Julia set iterates z^2 + c for fixed c and varying
starting z values. That is, the Mandelbrot set is in parameter space
(c-plane) while the Julia set is in dynamical or variable space