To iterate is to repeat the same process again and again. When used in reference to fractal calculations, an iteration usually implies using the results of one calculation as the input to the next. See also: recursion.
But what really is an iteration?
The equation used to calculate the Mandelbrot set is an extremely simple equation, involving nothing more complicated than addition and multiplication. What allows the infinite complexity of the Mandelbrot set to come out of such a simple equation (the actual equation is Z = Z * Z + C), is that the equation is iterated. That is, it isn't just calculated once per point, it is calculated many times. Each time the result of the previous iteration is fed into the new calculation. The basic idea is that if the value of Z always stays near zero, then the point is in the Mandelbrot set. If the value of Z heads off to infinity, then it isn't in the Mandelbrot set.
Recognizing when Z heads off to infinity is easy, but ensuring that it will never head off that way is considerably trickier. The only way to be certain would be to continue iterating the equation forever. Clearly this would not be practical, so instead we choose a maximum number of times to iterate the equation. The trick is to set it high enough so that all points heading off to infinity have time to declare their intentions, yet low enough so that pictures can be calculated in a reasonable time.