Fractals have always been a bit of a mystery for me.
What practical uses (beyond rendering to beautiful images) are there for fractals in the various programming problem domains? And please, don't just list areas that use them. I'm interested in specific algorithms and how fractals are used with those algorithms to solve something in practice. Please at least give a short description of the algorithm.
Absolutely computer graphics. It's not about generating beautiful abstract images, but realistic and not repeating landscapes. Read about Fractal Landscapes.
Perlin Noise, which might be considered a simple fractal is used in computer graphics everywhere. The author joked around that if he would patent it, he'd be a millionare now. Fractals are also used in animation and lossy image compression.