Kevin's answer gives a very good explanation of the role of mod $p$ reduction in the theory of Galois representations and automorphic forms. In this answer I will try to say something a little more technical about one way in which understanding the mod $p$ reduction of modular curves can be applied in arithmetic. The precise application that I will discuss is that of constructing congruences of modular forms.

If $f$ is a Hecke eigenform (of weight 2, to fix ideas), then associated to $f$ is a Galois representation $\rho_f:G_{\mathbb Q} \to GL_2(\overline{\mathbb Q}_{\ell})$ (for any prime $\ell$). Say the level
of $f$ is equal to $N p$, where $p$ is a prime not dividing $N$. One can ask: is there an eigenform $g$ of level $N$ such that $f \equiv g \bmod \ell$. (Here congruence means congruence
in $q$-expansions.) This is the question that Ribet solved in his famous Inventiones 100 paper (the paper which reduced FLT to Shimura--Taniyama).

Note that since $p$ is not in the level of $g$, the representation $\rho_g$ will be
unramified locally at $p$. (This comes from knowing that the modular curve of level $N$
has good reduction at $p$, since $p$ does not divide $N$ --- a first application of the theory of reduction of modular curves.) (If $p = \ell$ one must be more careful here,
but I will suppress this point.)

Thus if $f \equiv g \bmod \ell,$ so that $\rho_f$ and $\rho_g$ coincide mod $\ell$,
we see that $\rho_f$, when reduced mod $\ell$, must be unramified at $p$. So this
is a necessary condition for the existence of $g$.

It turns out (and Ribet proved) that (under some additional technical hypotheses) this
necessary condition is also sufficient. The way the argument goes is the following:
the modular curve of level $N p$ has semi-stable singular reduction: it is two smooth
curves (coming from level $N$) crossing each other a bunch of times (this is the contribution from the $p$-part of the level). Now the mod $\ell$ Galois representation
$\overline{\rho}_f$ (the reduction of $\rho_f$ mod $\ell$) is consructed out of
the $\ell$-torsion subgroup of the Picard group of this singular curve. Since it
is unramified at $p$, it can't be entirely explained by the singularities; some part of it
must be arising from the smooth curves, which are of level $N$. (If you like, this is an application of the a certain form of the criterion of Neron--Ogg--Shafarevic.) The Eichler--Shimura
relations then show that the system of Hecke eigenvalues attached to $f$, when reduced
mod $\ell$, must arise at level $N$: in other words, there *is* an eigenform $g$ of level $N$ that is congruent to $f$ mod $\ell$.

This is just one typical argument that uses a detailed knowledge of the good and bad reduction of modular curves in various situations. Since the Galois representation attached to modular forms are constructed geometrically from the modular curves, tools
like the Neron--Ogg--Shafarevic criterion, and variants thereof, show that there are
very close ties between the local properties of the Galois representations at a prime $p$,
and the reduction properties of modular curves mod $p$.