There is more than one definition to what is a fractal in general it is a structure which has self similarity, that is, it look the same when it is enlarged, like a cauliflower for example, each branch looks similar to the whole head.
There are many ways to create a fractal on a computer one way is by a recursive function of instructions such is the famous LOGO tree created by a V shape drawn on the tip of each previous V.
Another way to create much more "colorful" fractals is by iterating complex functions. The Julia and Mandelbrot Sets are of such nature. They are made with the simple function:
Zn+1=Zn2+A
where Zn=X+iY
and A is also a complex number.
In the Julia Set each point is given a different Z0 as a starting point while A is kept constant. The number of iterations, i.e. the first n for which Zn is farther than a predefined radius, determines the color. If the point converges the point is left black.
In the Mandelbrot Set the concept is the same except here Z0=0 and A changes for each point on the screen.
The Julia and Mendelbrot Sets are closely related and are acutely different "slices" of the same "plane" having one point in common. This is a transformation of the Julia set as A is changed. See how there is a connection between the place in the Mandelbrot Set (the white dot on the right) and the form of the Julia Set for that A.
Here are a few zooms into the Mandelbrout set.
Here is what happens when the Mandelbrot Set is started not from 0
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