Back when I wrote about the late Benoît Mandelbrot's fractal work, I really only covered the basics about the nature of fractals themselves.
Mathipedia recently alerted me to a documentary on fractals that gives a more complete picture.
The documentary is called Hunting the Hidden Dimension, and was an episode of NOVA back in 2008. It begins with the 19th century discovery of number sets with such unusual qualities, they were referred to as monsters, but regarded as little more than curiosities.
Once computers were developed, and maverick mathematicians like Mandelbrot were able to examine them, it was found that there was far more to fractals than anyone had expected. After explaining much about their nature, Hunting the Hidden Dimension examines the unexpected effects they've had on communications, medicine, entertainment, and much more. Without knowing it, you've probably taken advantage of the unique qualities of fractals without even knowing it!
If the video above ceases to work, you can also watch the complete documentary at the Hunting the Hidden Dimension homepage. The home page also has many more interactive features, such as playing with the scale and iterations of classic fractals.
Even if you don't have the Flash plug-in, features such as the interview with Benoît Mandelbrot himself are well worth checking out.