This applet is actually a graphical editor for something called an Iterated Function System (IFS). IFSs are used to describe linear fractals (a more correct term would be affine fractals and to be even more picky, IFSs need not be affine, although this applet implements a linear IFS model). A linear fractal is an image that is defined by copies of itself transformed by affine transformations. There are other types of fractals that contain self similar copies, such as the well known Mandelbrot set. The mathematics behind linear fractals is surprisingly simple, only a basic knowledge of linear transformations (matrices etc) is required.
The chaos game, sometimes called the random iteration algorithm, is one of several algorithms that can be used to generate linear fractals. It was invented by Michael Barnsley (Barnsley, M., Fractals Everywhere, Academic Press Inc., ISBN 0-12-079062-9, 1988). It is often described as a way of generating the Sierpinski Gasket, but the chaos game can generate any linear fractal.