This article is from the Fractal FAQ, by Ermel Stepp firstname.lastname@example.org with numerous contributions by others.
The basic algorithm is:
For each pixel c, start with z=0. Repeat z=z^2+c up to N times, exiting if
the magnitude of z gets large.
If you finish the loop, the point is probably inside the Mandelbrot set. If
you exit, the point is outside and can be colored according to how many
iterations were completed. You can exit if |z|>2, since if z gets this big it
will go to infinity. The maximum number of iterations, N, can be selected
as desired, for instance 100. Larger N will give sharper detail but take