Ok.. I nenglish (not my first language, so bear with me if I make a mistake:
First of all, when talking about a Z-transform, you ALMOST always mean a UNILATERAL Z-transform. So you can forget the others.
For electrical engineering it's nice to remember that laplace (as someone mentioned) deals with the DIFFERENTIAL equations and s-planes, where Z-transform deals with DIFFERENCE equations and the z-plane. However, the two are not identical, as the S-plane is "rectangular" where the Z-plane is "polar". In essence a Z-transform is the basics in a recursive digital filter !
Now if you're a student of electrical engineering, I guess that's about all you need to know to really understand it. The math behind it is a bit more complex to explain without an example. This link will give you a little more insight than the wiki:
http://www.ee.adfa.edu.au/Staff/a-lambe … 2Part4.pdfIf it's the MATH that's a problem, post again, and find out how many math buffs there truely ae in the BF2S forums (I'll bet a LOT)