Not a number theorist, but the wikipedia reads ok for me, so I’ll give an attempt. Answer based on the AMS’s Translated Math Monographs 240, by Kazuya Kato et. al…

A sample of the questions class field theory wants to address: a) Which primes p are the sum of 2 squares, p=a^{2} + b^{2}?

b) What about other formulae, say eg p=a^{2} +2b^{2}?

c) Consider a Galois extension. Take a prime ideal P in the smaller ring. For which primes does this ideal factor when we look at the larger ring?

d) When is the factorization square free (unramified)?

e) What’s the smallest cyclotomic extension that contains sqrt(M) for a given M?

If we look at the integers, you may already know the answers to several of these! And they all have something kinda magic in common. For (a), for example, the primes that are the sum of 2 squares are exactly those with p = 1 mod 4. For example, 5=2^2 + 1^2, yet 7 cannot be written as a sum of two squares. The answer to question (b) is similar! We can do it exactly when p=1,3 mod 8.

For ( c ), for concreteness let’s take the extension of the rationals Q to the rationals with a square root of -3, Q(sqrt(-3)). The prime ideal (7) factors as (7, 1-sqrt(-3)) (7, 1+sqrt(-3)) (a product of two distinct prime ideals; unramified), as do the ideals (13), (19), (31), and (37). But (5), (11), (17), (23) and (29) all don’t. Perhaps you notice a pattern: p=1 mod 3 ? factors. p=2 mod 3? doesn’t. There’s also a unique ramified prime, (3) = (sqrt(-3))^2. There will generally only be a finite number of ramified primes. Do a dozen more examples and you’ll notice a spooky pattern: the ramified primes seem to show up in the modulus (in this example, 3 was ramified and the factorization pattern works mod 3. If 7 and 23 are ramified, the factorization cases will work modulo 7*23=161). [Quadratic extensions are not special btw; the factorization of (p) in Q(zeta_5) (Q with a 5th root of 1) depends on p mod 5.]

On the face of it, why would modular arithmetic be the relevant condition? And why does the modulus seem to care about ramification?

A major result of Galois theory is that there’s a correspondence between subgroups of (Z/NZ)^* (integers modulo N under multiplication) and intermediate field extensions between Q and a cyclotomic extension Q(zeta_N). Prime ideal ramification and factoring can be stated in terms of this correspondence. Further, they show that every finite abelian extension of Q lives inside some Q(zeta_N). This result lets us explain all of (a)-(e). Generalizing it is one of the big motivations of class field theory. If we start not with Q, but with say Q(sqrt(-3)), what still holds? What is the right generalization of cyclotomic extensions and (Z/NZ)^*?

My understanding is that this program is quite successful. There’s a replacement for both that’s only somewhat more technical/tedious, and that gives similar results. One of the bigger successes is generalizing ‘reciprocity’ laws (the quadratic case is often taught in undergrad number theory; it’s about the surprising fact that p is a square mod q depends on if q is a square mod p).

Mm I don’t know, but it makes me want a cigarette.

I found the wikipedia article mostly incomprehensible but it says a few things. You are probably better off asking on MSE or Reddit, sorry to say. Wikipedia’s math reference desk has slowed down a lot in recent years though that’s possibly another place to try.