It is a triangle, with triangles connected to each side of that triangle, triangles attached to each side of that one, ad infinitum. The Mandelbrot set, considered by Mandelbrot himself to be even more fractal than fractals, is probably one of the most complicated mathematical problems in existence. The image that is produced when this is graphed is extremely complicated, and is similar, but not quite the same, on every scale. Lev D. Landau had a theory about fluid dynamics, that when energy is introduced into a system, each frequency is incompatible with the last and the system gets more and more complicated. Swinney and Gollub conducted an experiment with a rotating cylinder, and they observed Landau's theory break down, and the system become chaotic.
Phase space is a way of turning numbers into pictures.
For example, finding the relationship between three variables in three-demensional space. An attractor is the position a system will eventually turn to- for example, a pendulum is attracted to the center of the system, and eventually with air resistence, it will settle on that point. There are some new beliefs that go along with chaos theory.