A fractal is a geometric shape that is similiar to itself at different scales. More clearly, a fractal shape will look almost, or even exactly, the same no matter what size it is viewed at.
Fractal objects have several interesting properties. One of the most interesting is self-similarity. The Koch crystal is a good example of this. By adding triangles to the center of each side of a triangle, a more complex structure is obtained. If this process is repeated, we end up with successively more complicated, each of which, however, exhibit the same basic symmetry of the original triangle. The figure below shows how the Koch crystal is formed by repetition from a simple geometric figure.
Any fractal object exhibits self-similarity over many scales of observation. Another property of a fractal object is a lack of a well defined scale. A good example of this is clouds, which tend to look very similar no matter what their size.