A lot of people have said they don't know what it is. So here is an explanation of what it is, and how it is programmed:
The Mandelbrot set is a fractal. It is an infinitely complex structure. You can zoom in and in and in on it and it will never simplify. The way it is defined has a lot to do with complex numbers.
A complex number is a number with a 'real part' (just a normal number, like 1, -66.8, Pi etc.) added to an 'imaginary part' (this is a real number multiplied by the square root of minus one).
The "absolute value" or "size" of a complex number is the square root of the sum of the squares of the real and imaginary parts. | a+bi | = sqrt(a²+b²).
Each pixel on the screen can correspond to a complex number (i.e. the pixel at (13, 66) corresponds to 13+66i ,where 'i' is the square root of minus 1).
If you take a complex number C, say 1+1i, then add its square to it, then a new complex number Z is created. (in this case -11+7i which has an absolute value of 170). you can then square Z and add C to get a new value for Z. You can then square the new value of Z and add C to it again etc.
This can be repeated as an iteration, written as: Z => Z²+C. So C is a constant, and Z is a variable which initially starts at 0, and then changes every iteration. Here is the clever part:
The colour of each pixel on the screen is determined by the amount of iterations needed before the absolute value of Z becomes larger than a predetermined value, with the value of C depending on the
X and Y co-ordinates of the pixel. If, after many many iterations, the absolute value of Z is still below the predetermined value, then the pixel is coloured black.
This explanation probably sucks for anyone who doesn't already understand it., but at least I tried.