There are various options for making fractals. The most popular programs for making them are Apophysis and Ultra Fractal. The former is free, the latter is paid software.
As for the fractal cat/cheetah/various other animals fractalized, I don't believe these were generated with fractal programs at all, if they were at all, it wasn't purely done with the mathematical aspect of them.
There are however, options for using pictures in your fractals. With UF, I honestly don't know. With Apophysis(which can be found here:
http://xyrus02.deviantart.com/art/Apophysis-7X-134116371), after you get used to working with it, there is an Image plugin made by Xyrus02(the same guy who is updating apophysis) which is here:
http://xyrus02.deviantart.com/gallery/#/d2tschq
As for learning how to make fractals in the first place, deviantart has a multitude of resources in that regard. A good place to start if you're curious is
Fractal Resources. They have tutorials and other resources for various fractal related things.
And yes,
I am a fractal artist