The Buddhabrot is an alternate way of rendering the Mandelbrot set. Whereas conventional renderings color the points based on how many iterations it takes the point to diverge (if any), the Buddhabrot colors them based on how many times diverging numbers pass through that point. This causes interesting geometric patterns to occur.
The Buddhabrot is similar to a chaos game, in that points are chosen randomly to explore. While this causes the rendering to diverge initially, it tends to settle down based on the number of iterations specified. To see a visualization of the path numbers take on their quest to diverge, check out here.
Traditionally the Buddhabrot only counts the paths of points that escape, but you also render non-escaping points by checking the Non-Escaping box.