I don't have much time, Ohm, but the
wiki on the Z-transform is pretty thorough and does a good job of explaining it. It also discusses its relation to the Laplace and Fourier transforms. These are all useful tools for solving certain types of problems.
At the heart of such things, is a very important concept known as
convolution. The animations there speak volumes to explain the mathematics behind it. They show some relatively simple cases, but I think it's enough to get the general idea of the concept. Once you have this idea down, it's easier to move on to the transforms. As the first link I posted mentions, they deal with seeing how functions in the time domain react in the frequency domain, and vice-versa. Convolution is a messy process. Not terribly difficult (necessarily), but tedious. Using transforms allows one to use easier operators, such as straight multiplication, rather than convolution.