Someone once asked me "what are fractals? and how do you make those lovely images?". Not being much of a mathematician, and knowing that there are many webpages on the Internet that can answer the first question (I recommend the Wikipedia entry on fractals as a good place to start), I chose to try to answer the second question instead.
In order to make fractal art, you first need a computer and a fractal generation program. Of the latter, there are probably hundreds of different programs available, depending on what computer platform you have. Different programs do different things; many fractal generators are meant to address the more mathematical side of fractals, so while the end result of the program is still a fractal, it may not be very pretty, due to limited colour palette, or lack of anti-aliasing (smoothing).
The programs that I like to work with are KPT Fraxplorer, KPT Fraxflame, Ultra Fractal and Apophysis. Some other programs that I hear are good and may try out one day are Tierazon, Chaoscope, Flarium and Xenodream. A good basic (and free) program is Fractint. Once you've chosen a program, it's time to get down to business.
There are hundreds of different fractal formulas, with more being written every day. The ones I'll show here are two of the most widely used and most recognized; the Mandelbrot Set and the Julia Set.
For the purpose of this demonstration, I'm using Ultra Fractal. First you'll see the original fractal, with no colouring methods applied. Colouring methods are additional formulas applied on top of the basic fractal, which tell the program how to place the colour around the fractal. Then you will see how colouring methods can change or mask the basic look of the fractal. This can range anywhere from minimal adjustments to something so extreme that you cannot identify the original shape in the final image.
Click on the inline images to see a larger image.