A fractal is a kind of structure. It includes smaller versions of itself which can be explored to arbitrary levels of detail. It was discovered by Benoit B. Mandelbrot who coined the name "fractal" in 1975 from the Latin fractus or "to break".
You can explore fractals with a Fractal-walker.
The Mandelbrot set is the best known example of a fractal. The following images were created with the program Julia44 and show that a Mandelbrot set indeed includes smaller versions of it self.
See the page Free programs or my page with sources for the program Julia44