I think it is easiest to illustrate the role of the Langlands program (i.e. non-abelian class field theory) in answering this question by giving an example.
E.g. consider the Hilbert class field $K$ of $F := {\mathbb Q}(\sqrt{-23})$; this is a degree 3 abelian extension of $F$, and an $S_3$ extension of $\mathbb Q$. (It is the splitting field of the polynomial $x^3 - x - 1$.)
The 2-dimensional representation of $S_3$ thus gives a representation
$\rho:Gal(K/{\mathbb Q}) \hookrightarrow GL_2({\mathbb Q}).$
A prime $p$ splits in $K$ if and only if $Frob_p$ is the trivial conjugacy class
in $Gal(K{\mathbb Q})$, if and only if $\rho(Frob_p)$ is the identity matrix, if and only
if trace $\rho(Frob_p) = 2$. (EDIT: While $Frob_p$ is a 2-cycle, resp. 3-cycle, if and only if $\rho(Frob_p)$ has trace 0, resp. -1.)
Now we have the following reciprocity law for $\rho$: there is a modular form $f(q)$, in fact
a Hecke eigenform, of weight 1 and level 23, whose $p$th Hecke eigenvalue gives
the trace of $\rho(Frob_p)$. (This is due to Hecke; the reason that Hecke could handle
this case is that $\rho$ embeds $Gal(K/{\mathbb Q})$ as a dihedral
subgroup of $GL_2$, and so $\rho$ is in fact induced from an abelian character of the
index two subgroup $Gal(K/F)$.)
In this particular case, we have the following explicit formula:
$$f(q) = q \prod_{n=1}^{\infty}(1-q^n)(1-q^{23 n}).$$
If we expand out this product as $f(q) = \sum_{n = 1}^{\infty}a_n q^n,$
then we find that $trace \rho(Frob_p) = a_p$ (for $p \neq 23$),
and in particular, $p$ splits completely in $K$ if and only if $a_p = 2$.
(For example, you can check this way that the smallest split prime is $p = 59$;
this is related to the fact that $59 = 6^2 + 23 \cdot 1^2$.).
(EDIT: While $Frob_p$ has order $2$, resp. 3, if and only if $a_p =0$, resp. $-1$.)
So we obtain a description of the set of primes that split in $K$ in terms of
the modular form $f(q)$, or more precisely its Hecke eigenvalues (or what amounts
to the same thing, its $q$-expansion).
The Langlands program asserts that an analogous statement is true for any
Galois extension of number fields $E/F$ when one is given a continuous
representation $Gal(E/F) \hookrightarrow GL\_n(\mathbb C).$ This is known
when $n = 2$ and either the image of $\rho$ is solvable (Langlands--Tunnell) or $F = \mathbb Q$ and $\rho(\text{complex conjugation})$ is non-scalar (Khare--Wintenberger--Kisin).
In most other contexts it remains open.
In either the one or two dimensional case, there are two sides: Galois and automorphic.
Let me talk for a moment about the $n$-dimensional situation.
The Galois side involves studying continuous $n$-dimensional representations of the Galois group of
a number field $K$.
There are subtleties here about over what field these representations are defined, which I will return to. For now, let's imagine that they are representations into $GL_n(\mathbb C)$,
hence what are usually called Artin representations.
The automorphic side involves so-called automorphic representations of $GL_n(\mathbb A)$,
where $\mathbb A$ is the adele ring of $K$. These are irreduible representations which appear as Jordan--Holder factors in the
space of automorphic forms, which is, roughly speaking, the space of functions
on the quotient $GL_n(K) \backslash GL_n(\mathbb A)$. (To provide some orientation with regard to this notion: Note that, by Frobenius reciprocity,
any irreducible representation of a group $G$ appears in the space of functions on $G$ --- here I am ignoring topological issues --- and so any irreducible rep. of $GL_n(\mathbb A)$ will appear in the space of functions on $GL_n(\mathbb A)$. On the other hand, appearing in the space of functions on the quotient $GL_n(K)\backslash GL_n(\mathbb A)$ turns out to be a serious restriction --- most representations of $GL_n(\mathbb A)$ don't appear there.)
The idea --- roughly --- is that $n$-dimensional Galois representations will match
with automorphic representations. How does this happen?
Well, $n$-dimensional representations have traces of Frobenius elements,
so to each unramified prime we can attach a number. On the other hand, it turns out
that automorphic representations have essentially canonical generators, which can
be interpreted as Hecke eigenforms, and so (for primes not dividing the conductor
of the representation) we get a Hecke eigenvalue. The matching is given by the rule that traces of Frobenius elements should equal Hecke eigenvalues.
Let's consider the case of dimension $n = 1$:
First, any $1$-dim'l rep'n of the Galois group of $K$ factors through $G_K^{ab}$, while
the space of functions on the abelian group $K^{\times}\backslash \mathbb A^{\times}$
(which is the adelic quotient considered above in the case $n = 1$) is (by Fourier
theory for locally compact abelian groups, and speaking somewhat loosely) the sum of spaces spanned by characters, so that automorphic representations are just characters of
$K^{\times}\backslash \mathbb A^{\times}$.
So our goal is to match characters of $G_K^{ab}$ with characters of $K^{\times}\backslash
\mathbb A^{\times}$.
Now global class field theory actually does something stronger: it directly relates
$G_K^{ab}$ to $K^{\times}\backslash \mathbb A^{\times}$. In particular, finite order
characters of the latter (which are just Dirichlet characters in the case $K = \mathbb Q$)
correspond to finite order characters of the former, the $L$-functions match, and so on.
Now consider the case $n = 2$:
The quotient $GL_2(K)\backslash GL_2(\mathbb A)$ is not a group,
just a space (and this is the same for any $n \geq 2$). Also, there is no quotient
of the Galois group $G_K$ akin to $G_K^{ab}$ with the property that all $2$-dimensional reps. of $G_K$ factor precisely through that quotient.
In short, unlike in the case $n = 1$, we can't hope to find groups that will be related in some direct manner (unlike in the case $n = 1$, where class field theory gives a direct relation between $G_K^{ab}$ and $K^{\times}\backslash \mathbb A^{\times}$); the relation
really has to be made at the level of representations.
Also, when you consider the quotient $GL_2(K)\backslash GL_2(\mathbb A)$, it is much harder
(in fact, impossible) to separate the archimedean and non-archimedean primes. If you play around and follow up some of the references suggested by others, you will see (in the case $K = \mathbb Q$) that this quotient is related to the upper half-plane and congruence subgroups, and the generating vectors for automorphic representations are precisely primitive Hecke eigenforms (either holomorphic modular forms or else Maass forms). The archimedean prime contributes the upper half-plane, and the finite primes contribute the level of the congruence subgroup and the Hecke operators.
Thus the analogue of a Dirichlet character in the $2$-dimensional case is a Hecke eigenform.
(But it is better to regard the latter as corresponding more generally to idele class characters --- also called Hecke characters or Grossencharacters; these are not-necessarily-finite-order generalizations of Dirichlet characters.)
If you really want to study just Galois represenations with finite image (i.e. Artin representations), on the automorphic side (for $K = \mathbb Q$ and $n = 2$) you should restrict to weight one holomorphic modular forms and Maass forms with $\lambda = 1/4$. (The former are proved to correspond precisely to two-dimensional reps. for which complex conjugation has non-scalar image, while the latter are conjectured to correspond to two-dimensional reps. for which complex conjugation has scalar image.) The other Maass forms
are not supposed to correspond to any Galois representations (just as non-algebraic idele class characters --- those that are not type $A_0$ in Weil's terminology --- don't correspond to anything on the Galois side in class field theory), while higher weight holomorphic modular forms are supposed to corresond to compatible systems of two-dimensional $\ell$-adic Galois representations with infinite image (just as infinite order algebraic Hecke characters correspond to
compatible systems of one-dimensional $\ell$-adic Galois representations --- see Serre's bok on this topic).
As the preceding summary shows, the big picture here is pretty big, and the technical details are quite extensive and involved. There is the added complication that the proofs of what is known in the non-abelian case are very involved, and often use methods that don't play a role in the general story, but seem to be crucial for the arguments to go through. (E.g. Mellin transforms don't play any role in the general theory of attaching $L$-functions to automorphic representations, but in the case $n = 2$, the fact that you can pass from a modular form to its $L$-function via a Mellin transform is often useful and important.)
What I would suggest is that you try to learn a little about Hecke characters beyond the finite order case, perhaps from Weil's original article and also from Serre's book. (Silverman's treatment of complex multiplication elliptic curves may also help.) This will help you become familiar with the crucial idea of compatible families of $\ell$-adic Galois representations, and see how it relates to objects on the automorphic side, in the fundamental $n = 1$ case.
Then, to begin to grasp the $n = 2$ case, you will want to understand how modular forms
give compatible systems of two-dimensional $\ell$-adic representations. The general statement here is due to Deligne, and can be learnt (at least at first) as a black-box. The case of weight 2, which includes the case of elliptic curves over $\mathbb Q$, is easier, and it's possible to learn the whole story (other than proof of the modularity theorem itself) in a reasonable amount of time. The case of weight one is special, and is treated in a beautiful paper of Deligne and Serre. The converse here (that every two-dimensional
Artin rep'n of appropriate type comes from a weight one form) is at least as difficult as the modularity of elliptic curves, and so learning the statement should suffice at the beginning.
And now you confront the difficulties in learning non-abelian class field theory: already
in the two-dimensional case, the theory has some aspects that remain almost entirely conjectural,
namely the conjectured relationship between certain Maass forms and certain two-dimensional
Galois representations. So at this point, you have to choose whether you just want to get some sense of the big picture and the expected story in general (say by looking over Clozel's Ann Arbor article and the recent preprint of Buzzard and Gee), or to begin actually working in the area. If you choose the latter, then the big picture is useful, but only in a limited way: to make progress, it seems that one has to then start learning many techniques that don't seem to be essential to the story from the big picture point of view,
but which are currently the only known methods for making progress (depending on what direction you want to pursue, these include the trace formula, Shimura varieties, the deformation theory of Galois representations, the Taylor--Wiles method, ... ).
Best Answer
Here is an extremely naive answer to a case of my own question, which is surely obvious to experts. For me, even coming up with this silly version required quite a bit of conversation with Sugwoo Shin (who is of course blameless of any errors).
We give a description of the odd $GL_2$-Galoisian sets of level $N$ in terms of 'higher congruence conditions.' The point is to consider the $\mathbb{Q}$-Hecke algebra $H(N)$ determined by the Hecke operators acting on modular forms of weight 1, level $N$. The maximal ideals in $H(N)$ are in correspondence with Galois conjugacy classes of normalized new weight one eigenforms of level $N$. There is also a map $$p\mapsto T_p$$ from primes not dividing $N$ to $H(N)$.
Any given maximal ideal $m$ determines a Dirichlet character $\epsilon_m$, and one considers the set of primes $S(m)$ defined by the 'congruence conditions' $$(p,N)=1, \ \epsilon_m(p)=1, \ T_p\equiv 2 \ \ \mod \ m$$ These $S_m$ are exactly the odd $GL_2$ Galoisian sets of level $N$. With a bit more care, one should be able to give a similar description that doesn't refer to the level beforehand.
Of course this is no different from what I wrote before, but focussing on the Hecke algebra seems to allow a formulation that's rather analogous to the classical one. That is, one can forget about modular forms for a moment and examine sets of primes determined by congruences in the Hecke algebra.