Fractals are graphic representations of functions applied (iterated) over and over. The results of these mathematical functions are plotted, creating intricate designs.
You can see several different designs inside the fractal above. If you were able to zoom in on this image you would find that the design warps and changes forever. You could keep zooming in for infinity and find different shapes and designs.
Here is the same design, zoomed in:
Also, as you change mathematical functions you change the design of the fractal. Many programs also allow you to change colors. Here is a different design, created by changing the function:
Here are some pages I found helpful in learning about fractals:
This page was the best, I thought (Amazing Seattle Fractals!). It has tons of links and info:
I found plenty of software that creates fractals. Some programs are free, some cost money. Some are easy to use, others you have to be a mathematician to use.
Here is the program I am using to create mine.
If you fiddle with it for a few minutes you can create some great looking and large fractals, zoom in on them and save them in a bitmap format, which you can then import into Photoshop and do whatever you want with, adding or mixing other photos and creating compositions.
Fractal designs are pretty easy to do using the right software.