zn+1 = zn2 + c
where z0 = c
will never approach infinity for any n, that is, will never leave a circle around origo with a radius of 2.
Now, if you are bothered by complex numbers being non-intuitive, welcome to the club. Just view them as coordinates (x, y) in a coordinate system.
To perform this calculation, one needs to choose a "bail-out value" - a max value of n in the formula, to be able to terminate the calculation for those points that are inside the set - otherwise there would be no way to stop calculating. This is computationally demanding stuff, and one will have to choose this bail-out value carefully.
Or had to, at any rate. On today's computers you can probably set it to whatever you feel like.
This is an "in or not in" situation where the basic mathematical presentation is in the form of a b&w graph with the Mandelbrot set coloured black.
Rather boring stuff, in other words - unless you are sensible enough to be fascinated by the fact that this set will present infinite detail along the edges no matter how far you zoom into it, or by the fact that the edge of the set is single and unbroken. This edge, by the way, is of infinite length, even though the area of the mandelbrot set is finite
Now, the Mandelbrot graph can easily be turned into a colourful, psychedelic picture by simply assigning a colour to each point outside the Mandelbrot set proper, the colour chosen depending on how many iterations of the above formula it takes before z has left the r=2 circle. This is the kind of "Mandelbrot graph" you will usually see.
Actually, this does present some problems since you will have to define a colour table of a suitable size to match the actual set of n values in the actual subset of the graph that you are currently visualizing, and also depending on your current bail-out value. Each colouring model will present a subset in a different light.
Note: I've never tried using the Normalized Iteration Count Algorithm or anything like that to smooth out the colouring, since I consider it almost cheating. I've concentrated on improving the colour schemes without deviating from the normal, simple Escape Time algorithm.
In the menu you can find several applets to try out.
Change in formula
In the past, I've always used the formula that Benoît Mandelbrot himself published, "z -> z2 - c". However, the popular version of the formula nowadays is "z -> z2 + c", the only practical difference being the orientation across the y axis. Some of my images may still be made with the former formula. I'm in the process of tidying up my collection and converting old dumps but it's not ready yet.
I posted the following question on the Talk:Mandelbrot set page on Wikipedia in 2006, and received a good reply: