Places of an algebraic number field

I won’t help you with the ray-class group, but a place is either (a) a maximal ideal of the ring of integers of $K$ or (b) an equivalence class of archimedean metrics on the field.

For $\Bbb Q$, the places are the ordinary primes plus the “infinite” place of the standard absolute value you used in Calculus.

For $\Bbb Q(i)$, the places are the single prime $(1+i)$ dividing $(2)$; the primes $(a\pm bi)$ for $p=a^2+b^2$ a prime $\equiv1\pmod4$; and the ordinary primes $\equiv3\pmod4$. Again there’s only one infinite place, the familiar absolute value on $\Bbb C$.

For $\Bbb Q(\sqrt2\,)$, there again is only one ramified prime $(\sqrt2\,)$, and the other natural primes split or don’t according as $p\equiv\pm1\pmod8$ for the split ones; or $p\equiv\pm3\pmod8$ for the ones that remain prime. For instance, since $7$ is of form $a^2-2n^2$ for $a=3$, $b=1$, we may write $(7)=(3+\sqrt2\,)(3-\sqrt2\,)$. So there are two places of $K=\Bbb Q(\sqrt2\,)$ above $(7)$, while $(3)$ remains prime in $K$, and there’s just the single place $(3)$ above three. The archimedean primes are more fun. You can embed $K$ into $\Bbb R$ in two ways, by sending a particular $\lambda$ for which $\lambda^2=2$ to the positive or the negative square root of $2\in\Bbb R$. So that gives you two inequivalent archimedean metrics on $\Bbb Q(\sqrt2\,)$.

In general, if $[K:\Bbb Q]=n$, there will be at most $n$ places of $K$ above any place of $\Bbb Q$. Hope this helps.

All the technical semantics are due to the infinite primes, or infinite places, or achimedean primes (or whatever) when defining an infinite prime of an arbitrary number field K.

For any embedding $\sigma$ of K into an algebraic closure, the infinite prime $P_{\sigma}$ is just a formal symbol attached to $\sigma$. A problem arises when considering an extension $L/K$ s.t. $\sigma$ embeds K into $\mathbf R$, but extends to 2 conjugate complex embeddings $\tau, \tau '$ of L into $\mathbf C$. In this case one defines the infinite prime $P_{\tau}$ over $P_{\sigma}$ as being the formal symbol $P_{\tau}=P_{\tau'}$, and one says $P_{\tau}$ is ramified over $P_{\sigma}$. The general statements about ideals are thus extended coherently if we look at the completions ($\mathbf R$ and $\mathbf C$ here), but incoherently if we consider splitting phenomena (e.g. a finite prime ideal P of K which splits as Q.Q' in L, in which case Q is unramified above P in the classical parlance).

A modulus of K is by definition a formal product $M=M_0.M_{\infty}$, where $M_0$ is an ordinary ideal and $M_{\infty}$ is a formal product of real infinite primes of K (all the infinite primes are raised to the first power here.) Then the ray class-field $K_{M}$ is the maximal abelian extension of K which is unramified outside $M$, and Takagi's existence theorem does need modulii. An excellent account of all this, with many examples, is §7 of D. Garbanati, "CFT summarized", Rocky Mountain J. of Math., 11,2(1981), 195-225.