Let me ask a few simple concrete questions (whose answers I’m sure are well known) to motivate my study of class field theory:
What is the maximal abelian unramified extension of $\mathbb{Q}[i]$? (I guess this is called the Hilbert class field).
What is the maximal abelian extension of $\mathbb{Q}[i]$ unramified everywhere except at a prime $p \in \mathbb{Z}$?
Same questions for $\mathbb{Q}[\sqrt{-5}]$ which does not have class number one.
$\Bbb Q(\sqrt{-5})$ has class number two, so its Hilbert class field is a quadratic extension. That quadratic extension is $\Bbb Q(i,\sqrt5)$.