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.
Error diffusion along a Hilbert curve.
It's a simple idea - suppose that you convert an image to a 0-1 black & white bitmap. Converting a 55% brightness pixel to white yields a +45% error. Instead of just forgetting it, you keep the 45% to take into account when processing the next pixel. Suppose its value is 80%. Normally it would be converted to white, but a neighboring pixel is too bright, so taking the +45% error into account, you convert it to black (80%-45%=35%), keeping a -35% error to be spread into next pixels.
This way a 75% gray area will have white/black pixel ratio close to 75/25, which is good. But if you process the pixels left-to-right, the error only spreads in one direction, which yields worse looking images. Enter space-filling curves. Processing the pixels along a Hilbert curve gets good locality of the error spread. More here, with pictures.