Categories of mathematics

I first refer you here, to the math subject classification system of the American Mathematical Society. I also refer you Arxiv's Math subject classification system. These are the two major systems that I use and that I refer to when classifying or looking for mathematics. As for the categories - these are often made the way they are due to historical events or interpretations.

In reference to the distinctions between 'first' and 'second' level math, and so on: I think that Arturo was basing these on necessary prerequisites. For example, one can take a first class on Algebra, Geometry, Elementary Number Theory, Real Analysis, or Topology without having taken any of the others. Of course, one might argue that there are many interconnections and that one would benefit from knowing algebra before learning number theory, or topology before real analysis, etc. I think this is true, but that it misses the point: it's not necessary at first.

On the other hand, algebraic number theory, algebraic topology, analytic geometry, etc (to directly quote your quote of Arturo) all require multiple previous topics, i.e. some mixture of topology, number theory, algebra, geometry, analysis, etc.

Based on the AMS classification scheme (linked in mixedmath's answer) the Mathematical Atlas has visual representations of overlaps, connections between fields, etc. I encourage you to explore the math-map link, and other links in right upper corner of the link I'm providing:

http://www.math.niu.edu/~rusin/known-math/index/index.html

http://www.math.niu.edu/~rusin/known-math/index/mathmap.html

I found it helpful when I first encountered it. It elaborates a bit on the indexing/categories given by the AMS (American Mathematical Society).

Enjoy exploring! (It can be overwhelming to realize just how expansive the field of mathematics is, so take your time!)