Fractal shapes are shapes that resemble themselves when you look at them closer or further away.
A common example from nature is the coastline: you can look at it from the sky, from your height, up close,
with a microscope, and you will the same kinds of shapes from each perspective.
Fractals typically have other interesting mathematical properties as well: for example some fractals are lines but if you zoom in enough the 1 dimensional line will fill up the two dimensional plane (or part of it). In addition, mathematical fractals (and ones in nature!) are often beautiful.
Perhaps the most popular fractal is the Mandelbrot set, named after the mathematician who discovered it. You can read more it about in Wikipedia, or just search on the Web.
This page lets you explore the Mandelbrot set (the parts in black). Click to re-center the image. Click and drag to select a new portion to view. Alt-double click to zoom in at the point you clicked on. Shift-double click to zoom out at the point you clicked on.
This example is based on the example on Paul Bourke's site, which has lots of other cool fractals as well.