Caveat: in order to give you an overview, I've been vague/sloppy in several places.
Well the basic link to representation theory is that modular forms (and automorphic forms) can be viewed as functions in representation spaces of reductive groups. What I mean is the following: take for example a modular form, i.e. a function $f$ on the upper-half plane satisfying certain conditions. Since the upper-half plane is a quotient of $G=\mathrm{GL}(2,\mathbf{R})$, you can pull $f$ back to a function on $G$ (technically you massage it a bit, but this is the main idea) which will be invariant under a discrete subgroup $\Gamma$. Functions that look like this are called automorphic forms on $G$. The space all automorphic forms on $G$ is a representation of $G$ (via the right regular represenation, i.e. $(gf)(x)=f(xg)$). Basically, any irreducible subrepresentation of the space of automorphic forms is what is called an automorphic representation of $G$. So, modular forms can be viewed as certain vectors in certain (generally infinite-dimensional) representations of $G$. In this context, one can define the Hecke algebra of $G$ as the complex-valued $C^\infty$ functions on $G$ with compact support viewed as a ring under convolution. This is a substitute for the group ring that occurs in the representation theory of finite groups, i.e. the (possibly infinite-dimensional) group representations of $G$ should correspond to the (possibly infinite-dimensional) algebra representations of its Hecke algebra. This type of stuff is the basic connection of modular forms to representation theory and it goes back at least to Gelfand–Graev–Piatestkii-Shapiro's Representation theory and automorphic functions. You can replace $G$ with a general reductive group.
To get to more advanced stuff, you need to start viewing modular forms not just as functions on $\mathrm{GL}(2,\mathbf{R})$ but rather on $\mathrm{GL}(2,\mathbf{A})$, where $\mathbf{A}$ are the adeles of $\mathbf{Q}$. This is a "restricted direct product" of $\mathrm{GL}(2,\mathbf{R})$ and $\mathrm{GL}(2,\mathbf{Q}_p)$ for all primes $p$. Again you can define a Hecke algebra. It will break up into a "restricted tensor product" of the local Hecke algebras as $H=\otimes_v^\prime H_v$ where $v$ runs over all primes $p$ and $\infty$ ($\infty$ is the infinite prime and corresponds to $\mathbf{R}$). For a prime $p$, $H_p$ is the space of locally constant compact support complex-valued functions on the double-coset space $K\backslash\mathrm{GL}(2,\mathbf{Q}_p)/K$ where $K$ is the maximal compact subgroup $\mathrm{GL}(2,\mathbf{Z}_p)$. If you take something like the characteristic function of the double coset $KA_pK$ where $A_p$ is the matrix with $p$ and $1$ down the diagonal, and look at how to acts on a modular form you'll see that this is the Hecke operator $T_p$.
Then there's the connection with number theory. This is mostly encompassed under the phrase "Langlands program" and is a significantly more complicated beast than the above stuff. At least part of this started with Langlands classification of the admissible representation of real reductive groups. He noticed that he could phrase the parametrization of the admissible representations say of $\mathrm{GL}(n,\mathbf{R})$ in a way that made sense for $\mathrm{GL}(n,\mathbf{Q}_p)$. This sets up a (conjectural, though known now for $\mathrm{GL}(n)$) correspondence between admissible representations of $\mathrm{GL}(n,\mathbf{Q}_p)$ and certain $n$-dimensional representations of a group that's related to the absolute Galois group of $\mathbf{Q}_p$ (the Weil–Deligne group). This is called the Local Langlands Correspondence. The Global Langlands Correspondence is that a similar kind of relation holds between automorphic representations of $\mathrm{GL}(n,\mathbf{A})$ and $n$-dimensional representations of some group related to Galois group (the conjectural Langlands group). These correspondences should be nice in that things that happen on one side should correspond to things happening on the other. This fits into another part of the Langlands program which is the functoriality conjectures (really the correspondences are special cases). Basically, if you have two reductive groups $G$ and $H$ and a certain type of map from one to the other, then you should be able to transfer automorphic representations from one to the other. From this view point, the algebraic geometry side of the picture enters simply as the source for proving instances of the Langlands conjectures. Pretty much the only way to take an automorphic representation and prove that it has an associated Galois representation is to construct a geometric object whose cohomology has both an action of the Hecke algebra and the Galois group and decompose it into pieces and pick out the one you want.
As for suggestions on what to read, I found Gelbart's book Automorphic forms on adele groups pretty readable. This will get you through some of what I've written in the first two paragraphs for the group $\mathrm{GL}(2)$. The most comprehensive reference is the Corvallis proceedings available freely at ams.org. To get into the Langlands program there's the book an introduction to the Langlands program (google books) you could look at. It's really a vast subject and I didn't learn from any one or few sources. But hopefully what I've written has helped you out a bit. I think I need to go to bed now. G'night.
For abelian number fields, the Dedekind zeta function factors into Dirichlet $L$-functions. Hecke characters are one-dimensional representations. For possible generalizations to non-abelian field extensions, you will require higher dimensional Galois representations 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. Similar things happen for modular forms of weight one (Hecke 1925). In general, automorphic representations and Galois representations are expected to be in a certain 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 representation, and the later was then shown to correspond to an automorphic form.
Best Answer
Galois representations have to come from somewhere.
If you are not interested in learning about modular forms and automorphic forms at this point, the other best source of representations are elliptic curves. You can find a gentle introduction in Silverman's book "The Arithmetic of Elliptic Curves", particularly the chapter on Tate modules.
As an alternative, for a completely "algebraic number-theoretic" approach, you might want to learn about complex Galois representations and their Artin L-functions. This is covered for example of Neukirch's book on ANT [Chapter VII] or Lang's [Chapters VIII and XII]. There's a lot of interesting and accesible number theory around it. That said, this alone won't get you too far in the way of understanding the big picture of Galois representations.
A nice general overview which doesn't jump directly into automorphicity issues is Richard Taylor's survey "Galois Representations", but it gets technical very quick.
You also might want to keep in mind the more advanced references in Emerton's great answer to this question.
There's also plenty of introductory text to modular forms. "A First Course in Modular Forms" has alredy been mentioned in the comments. The classic "A Course in Arithmetic" by Serre is probably the best general textbook on number theory that covers the basic of modular forms.