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"Fractal" is a contraction of "fractional" referring to the fact that fractals will often have dimension that are not integers. I remember reading somewhere that a fractal is a shape that has a Hausdorff dimension strictly greater than its topological dimension. I don't know what that means, though. So that isn't very helpful.

Coastlines, the outlines of leaves and many other natural phenomena are said to be fractal. Even mathematically generated fractals like the Mandelbrot set are quite often studied empirically by testing their shape at certain points, and looking at a set associated with that point, called the Julia set.

Some fractals are self-similar; this means that parts of them look exactly like smaller parts of them scaled appropriately. The (middle third) Cantor set and the Sierpinski Gasket for instance are self similar. Not all fractals are strictly self similar. The Mandelbrot set is almost, but not quite self similar. There are parts of it that look very much like other parts of it, but not exactly the same.

Mathematically generated fractals can be astoundingly beautiful if skillfully constructed and coloured appropriately. I quite often use a program called Apophysis to create a "flame fractal" to use as my background.

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