So, what is the Mandelbrot Set?
First of all what is a set. In the classic definition of Georg Cantor it is a "Gathering together into a whole of definite distinct objects of our perception or of our thought." and it is one of the most fundamental concepts in mathematics. In this case, the Mandelbrot Set is a collection of complex numbers which have the property that when a certain operation is iterated on a member of the set, the sequence of complex numbers that is generated always stays finite. For complex numbers outside the set, the iteration eventually diverges away to infinite values. (An iterated operation means that we apply a certain operation to the first number we thought of, get a result, apply the same operation to that number, get a result, apply the same operation to that number ... and so on and on and on. The behaviour of such sequences is connected to whether we can use mathematical equations to predict the future, e.g. the weather, by jumping forwards in small time-steps.)
A complex number is made of two parts (usually called the real and the imaginary parts - these are just arbitrary labels, don't get hung up on them) and you can, if you wish just think of this special type of number as a pair of ordinary numbers that always travel together, say (3.23,-4.123), which you work with according to certain well-defined rules. Here, the first of the par is the "real" part and the second the "imaginary" part. We can therefore, if we wish, think of this as a coordinate pair and represent a complex number as a point on a 2D plane, just by using the the real part to define the X-coordinate the imaginary part to fix the Y-coordinate. You have to be able add, subtract, multiply and divide numbers and complex numbers come with a set of rules on how we do this. It happens that the easiest way to remember the rules is to think of the number with the imaginary part multiplied by the square root of -1. So we might write our complex number as 3.23+4.123i (where "i" here means the square root of -1). There is, of course, no real number that when multiplied by itself gives -1 and hence complex numbers sometimes give a lot of grief to novice mathematicians: how can you do valid calculations with a number that does not exist? Don't worry! Mathematicians have proved that if you use the rules properly you will never generate a contradiction and complex numbers turn out to be fantastically useful for solving many difficult problems. (We could not calculate the physics of atoms without complex numbers.) In fact, mathematicians sometimes talk about the "magic" of complex numbers because of the way they make difficult problems easy.
In illustrations of the Mandelbrot Set it is often conventionally the dark region of the plane which represents the Set itself (this is where iteration of the associated complex number does not diverge away to infinity). Colours are often used in the region outside the set boundary to encode the number of iterations required before it is clear that we are on a diverging track.
The mathematically interesting features of the Set are that a high degree of complexity arises from an exceptionally simple operation: we take our original complex number, c, and apply the iteration zn+1 = (zn)2+c, where our first iteration is with zo=0 and so z1 = 0 + c2 , We then count the number of iterations until the distance of z from the origin reaches a value that allows us to conclude that c is definitely not in the Mandelbrot set, or we reach an upper limit on the number of iterations we are prepared to tolerate, say 200 or so (in which case we conclude that c is probably in the Mandelbrot set). Having counted the number of iterations to diverge to an outer boundary, we use that the determine a colour and apply that colour to the point on the image plane that represents c. The image on this page, if you zoom in enough, always has a black filament running through the most complex areas. The images also have the property of self-similarity - zoom into small regions and they look just like larger regions. Zoom in again, as much as you like and you will get the same characteristics.
The boundary of the set - on one side in, on the other out - is fractal. However much you zoom in it still has a highly irregular shape. The Set is, however a connected region, from any point in the Set you can make a track to any other point in the Set always staying within the boundary.