The Mandelbrot set is a famous fractal set. This is an Ikeda attractor. It is the set to which the orbit of any point in the plane is attracted if we keep iterating a particular map from the plane to itself. (In this case the Ikeda map, for a particular choice of the two parameters that determine the Ikeda family.) The attractor itself is a fractal. The Ikeda map also shows chaotic behavior: if you start with a point on the attractor, then iterating the map will make it hop to other points in a seemingly haphazard way; if you start with any two points, however close to each other, then their iterates will go very separate ways later.
This extreme sensitivity to small deviations in the description of the initial state, which can amplify into enormous differences later, is what characterizes chaos. We encounter it in many dynamical processes; the presence of chaos is what makes weather prediction hard.
Figure credit: Glenn Elert.
URL for this and other pictures of blowups of the Mandelbrot set with coarser scales, or for other pictures of Julia sets: http://hypertextbook.com/chaos/eyecandy