I'll be straying outside of my expertise here, so the following probably contains some errors. I'm just adding my perspective as someone who has (at one time at least) thought about some of this stuff. Hopefully this will help foster some discussion.

1) Modular curves are locally symmetric spaces, meaning they are of the form $\Gamma\backslash G/K$ where $G$ is (for simplicity) a semi-simple Lie group (e.g. $SL_2({\bf R}$), $K$ a maximal compact subgroup (e.g. $SO(2)$), and $\Gamma$ a discrete subgroup of finite index in $G({\bf Z})$ (e.g. $SL_2({\bf Z})$) (an "arithmetic" subgroup). Margulis' work on rigidity of lattices in Lie groups implies that unless $G$ is $SO(1,n)$ or $SU(1,n)$, $\Gamma$ is a congruence subgroup of $G({\bf Z})$ (a congruence subgroup is an arithmetic subgroup containing the kernel of the reduction map $G({\bf Z})\rightarrow G({\bf Z}/N{\bf Z})$). Note that $SO(1,2)\simeq SL_2({\bf R})$, $SO(1,3)\simeq SL_2({\bf C})$.

In my experience, people typically only talk about hyperbolic (or symmetric) spaces and fuchsian groups in passing because there is more information available in explicitly using $G$ and $\Gamma$ (So they may start out discussing e.g. a quotient of hyperbolic n-space by a discrete subgroup, but they prove things using that $H^n$ is really $SO(n,1)/SO(n)$ and $\Gamma$ is really a subgroup of $SO_{n,1}({\bf Z})$).

When $\Gamma$ is a congruence subgroup of $G({\bf Z})$, you can think of $\Gamma \backslash G/K$ as $G({\bf Q})\backslash G({\bf A})/K\cdot K_f$, where $K_f$ is a compact open subgroup of $G({\bf A_{\rm f}})$ This tends to be simpler to work with, since $G({\bf Q})$ has a simpler structure than $\Gamma$ (algebraic groups over fields instead of rings). Note that strange things can happen with noncongruence subgroups (e.g. there might not be any cusp forms).

Automorphic forms are certain functions on $\Gamma\backslash G$. Modular forms are classically defined on $G/K$, but you can do a little transform-and-lift to get them as certain "holomorphic" automorphic forms. With automorphic forms, the power of representation theory enters, and you have automorphic $L$-functions (and Hecke operators).

2) Modular curves are Shimura varieties, meaning (kind of) that $G/K$ has a complex structure, and so (after some work) $\Gamma \backslash G/K$ is an algebraic variety. More work shows that the Shimura variety is defined over a number field (the canonical model of the variety over the reflex field).

Thus you can attach to it a Hasse-Weil zeta function, which naturally factors as an alternating product of $L$-functions attached to the cohomology groups of the Shimura variety. These are supposed to be automorphic. In the modular case, the Eichler-Shimura relation makes this connection pretty simple to prove, but it is not known in general, and is really hard. (I'm not sure what the state of the art is. I'm pretty sure Hilbert modular varieties ($GL_2$ over a totally real field), Picard modular surfaces ($SU(2,1)$), and the next-simplest Siegel modular variety ($GSp_4$, thinking of $GL_2$ as $GSp_2$) are known. I think more general unitary groups are known, but I can't pin down exact statements).

The cohomology groups carry an action by $G({\bf A}_f)$, which gives rise to an action via Hecke operators (thinking of Hecke operators as members of the group algebra for $G({\bf A}_f)$). There are simpler ways to see this. The etale cohomology groups also carry an action by the absolute Galois group of the reflex field, which gives rise to $\ell$-adic Galois representations.

Modular/automorphic forms are sections of ("automorphic") vector bundles on $\Gamma \backslash G/K$. The algebraic structure on the Shimura variety has consequences for automorphic forms e.g. in terms of rationality of Fourier coefficients and special values of $L$-functions.