Question

(In the context of Maxwell's equations:) what is divergence, and what is different about the dot product and 'x' product of them(in the equations)?

Answer 1

These are concepts in vector analysis and vector calculus. Are you at all familiar with these areas?

Answer 2

Not very well: can you recommend any books about introductory vector calculus?

Answer 3

http://www.mecmath.net/calc3book.pdf
That is a free textbook that I've heard good things about. I haven't ever read it through myself, though, so take that with a grain of salt.

To be an effective physicist, you need an extremely strong grounding in the mathematics that underlies the physical theories that you're working with. In nonrelativistic electrodynamics, that tends to be 3-dimensional vector calculus. But it's also important to see the mathematics applied to real physical problems.... for that I'd recommend Griffith's Introduction to Electrodynamics.