Consequences of the Langlands program

There are many applications of the Langlands program to number theory; this is why so many top-level researchers in number theory are focusing their attention on it.

One such application (proved six or so years ago by Clozel, Harris, and Taylor) is the Sato--Tate conjecture, which describes rather precisely the deviation of the number of mod $p$ points on a fixed elliptic curve $E$, as the prime $p$ varies, from the "expected value" of $1 + p$.

Further progress in the Langlands program would give rise to analogous distribution results for other Diophantine equations. (The key input is the analytic properties of the $L$-functions mentioned in Jeremy's answer.)

At a slightly more abstract level, one can think of the Langlands program as providing a classification of Diophantine equations in terms of automorphic forms.

At a more concrete level, it is anticipated that such a classification will be a crucial input to the problem of developing general results on solving Diophantine equations. (E.g. all results in the direction of the Birch--Swinnerton-Dyer conjecture take as input the modularity of the elliptic curve under investigation.)

In very general terms the Langlands correspondence implies that the L-functions of algebraic varieties are automorphic, and therefore they have analytic continuations and functional equations generalizing the properties of Riemann's Zeta Function.

The analytic consequence of the fact that "all elliptic curves over the rationals are modular" is that the Hasse-Weil $L$-function of such curves has an analytic continuation and functional equation.

I suppose this counts as an "application" in number theory.