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.
If you have already learnt group theory, I may suggest you to go through
the book 'theory of algebraic numbers' by Pollard & Diamond. It's a really
good treatise to start off. You don't even need to know the definition of
ring to read this book. Everything is given there in a very well setup.
After having finished that book, you may pay a look at 'A Classical
Introduction to Modern Number Theory' by Ireland & Rosen.
Best Answer
Here's a flavor of the kind of concepts that are introduced in the first three chapters of Tom Apostol's Introduction to Analytic Number Theory (Undergraduate Texts in Mathematics, Springer):
Divisibility, greatest common divisors, prime numbers, Fundamental Theorem of Arithmetic, Euclidean algorithm. Series of reciprocals of the primes.
Arithmetical Functions, Dirichlet product, Möbius inversion formula, Formal power series, Bell series of an arithmetical function, Derivatives of arithmetical functions, Selberg identity.
Big oh notation, asymptotic equality of functions, averages of arithmetical functions.
Chapter 4 deals with the distribution of primes. Congruences are not introduced until Chapter 5; some results on finite abelian groups and their characters occupy Chapter 6. Their main purpose is to tackle Dirichlet's Theorem of primes in arithmetic progressions on Chapter 7. The book continues after that.
By contrast, William LeVeque's Fundamentals of Number Theory (Dover), which leans more towards the algebraic side, we have that Section 1.1 is titled What is number theory?, followed immediately by section 1.2, Algebraic properties of the set of integers. The first major chapter is Chapter 2, dealing with unique factorization and the GCD, much like Apostol (but not dealing at all with a series, whereas Apostol already has a series in that first chapter). Chapter 3 deals with congruences, Chapter 4 with primitive roots and the group of units modulo $m$, Chapter 5 with quadratic residues and quadratic reciprocity, and not until Chapter 6 are arithmetical functions introduced.
Serge Lang's Algebraic Number Theory (Graduate Texts in Mathematics, Springer) does not even speak about arithmetical functions. We go directly to unique factorization of ideals in Dedekind domains. Though it does have a part entitled Analytic Theory (Part 3, comprising chapters XIII through XVII), they are concerned with the zeta function, Tate's Thesis, the density of primes, and the Brauer-Siegel Theorem.
Questions like "What is the probability that two 'random' integers are relatively prime?" (Answer: $\frac{6}{\pi^2}$) "What is the average order of the divisor function?" (Answer, $$ \sum_{n\leq x} = x\log x + (2C-1)x + O(\sqrt{x})$$ where $C$ is Euler's constant); these are the province of Analytic Number Theory. Most of these questions cannot be reasonably answered (if at all) with the standard tools of algebraic number theory (Galois theory, extensions of $\mathbb{Q}$, rings of integers, etc.) Just like Lang's book does not even mention arithmetic functions, Apostol's does not even mention Galois.
Though both seek to answer questions about the properties of the positive integers, the kind of questions that Analytic Number Theory and Algebraic Number Theory ask have a distinct flavor, with the former concerned with "limiting" questions while the latter is not, and the latter being concerned with "structural" questions while the former not so much. And the kinds of tools that each reaches for is likewise different. Thus, while Algebraic Number Theory can tell you that there are infinitely many primes of each of the forms $4n+1$ and $4n+3$, and Algebraic Number Theory can even tell you that they each occur with density $\frac{1}{2}$ among all primes (via Cebotarev's Density Theorem), Algebraic Number Theory would have a hard time proving that "running totals" change leads infinitely often (that is, that there are infinitely many integers $N$s such that the number of positive primes less than $N$ that are congruent to $1$ mod $4$ is larger than the number of primes less than $N$ that are congruent to $3$ mod $4$, and that there are infinitely many $M$s such that the number of positive primes less than $M$ congruent to $3$ mod $4$ is larger than the number of positive primes less than $N$ congruent to $1$ mod $4$). In fact, I'm not aware of any "algebraic" proof of this fact.