For abelian number fieldfields, the Dedekind zeta function factors into Dirichlet $L$-functions. Hecke character characters are one-dimensional representations. For possible generalizations to non-abelian field extensionextensions, you will require higher dimensional Galois representationrepresentations and Artin $L$ functions.
(Maass 1949) Maass wave forms of eigenvalue $1/4$ correspond to 2-dimensional Galois representations, see Bump "Autom.reps...." Chapter 1.9. Same think Similar things happen for modular forms of weight one (Hecke 1925). In general, automorphic representation representations and Galois representations are expected to be in a certain correspondance correspondence (the $L$ functions and the root numbers should be the same).
A possible conceptual explanation for the importance of Galois representations delivers the Tannaka-Krein theorem. Roughly, this states that knowing the representation theory is equivalent to knowing the group. The group you want to understand is the absolute Galois groups (with a profinite topology) via its Galois representations, and understand the Galois representation via automorphic forms.
Perhaps one famous example is the Taniyama Shimura conjecture and consequently Fermat's last theorem: A certain construction with the elliptic curve gave a Galois representationsrepresentation, and this the later was then shown to correspond to an automorphic form.