Oh God, there is so much involved in each one. Topology is the study of a space and number theory is analysis of properties of numbers.
...Wow.That was absolutely a terrible explanation. I'm sure WikiPedia would be much more helpful in answering.
EDITs:
http://en.wikipedia.org/wiki/Topology
http://en.wikipedia.org/wiki/Number_theory
I can safely say that I'm not too terribly knowledgeable on everything involved in each category, but then again, I don't think I could define algebra, calculus, or geometry in a few sentences either. Of the bits and pieces I've read about course offerings at MIT and ASU, I believe these are the two areas of mathematics that most interest me.
Topology is quite similar to geometry, except it doesn't concern finite distances so much as it focuses on shapes, ratios, and functions that define the shapes. The book
The Poincaré Conjecture discusses topology quite exclusively as it pertains to our ever expanding universe.
Number Theory interests me because I have a reasonably difficult time applying mathematical concepts. It just wasn't my thing. We discuss kinematics in physics and instantly my buddy Tom can take these simple algebraic equations and translate all of it into complex spacial vector calculus. Number Theory looks at things like the prime number theorem, the number zero, pi,
e, Phi/phi, etc., and breaks it all down. Importance of numbers and their applications in other indefinite mathematics intrigues me.
I'm on the lookout for a couple of classes on sacred geometry also, but I haven't found any...yet.