TITLE: Langlands program vs Shimura-Taniyama-Weil conjecture
QUESTION [7 upvotes]: Edward Frenkel said that "we can  see Langlands program as a generalization of Shimura-Taniyama-Weil conjecture in the case of elliptic curves" 
I hope I'm not distorting his phrase, can someone explain what that means. Lets say that I'm little bit familiar with the ingredients used in both conjectures, Galois representations, elliptic curves,...

REPLY [6 votes]: To expand on zeno's answer, a Langlands-type formulation of the modularity conjecture would be:

(Taniyama-Shimura) $L(E,s)=L(f,s)$

Here $L(E,s)$ is the Hasse-Weil L-function of an elliptic curve $E$ over $\mathbb{Q}$, and $L(f,s)$ is the L-function of a modular form $f$ of weight 2 with integral coefficients. In fact you have a complete correspondence: all elliptic curves (over $\mathbb{Q}$) are modular in that sense, and all newforms (of that type) come from elliptic curves.
A "Langlands generalization" of this result is:

(Langlands) $L(V,s)=L(\pi,s)$

Here $L(V,s)$ is the Hasse-Weil L-function of any algebraic variety $V$ over an arbitrary number field, and $L(\pi,s)$ the L-function of some automorphic form $\pi$ over $\mathrm{GL}_n(\mathbb{A})$.
This is a theorem in some cases (other than Taniyama-Shimura), for example abelian varieties over $\mathbb{Q}$, elliptic curves over real quadratic fields and elliptic curves with complex multiplication.
You can further generalize:

(Langlands) $L(M,s)=L(\pi,s)$

With $L(M,s)$ the L-function of a motive. This of course includes the case above, Artin L-functions (non-abelian class field theory) and all the other Galois representations that should be automorphic ($l$-adic, mod p).
In the other direction, the Langlands conjectures restricted to the the case of an elliptic curve over $\mathbb{Q}$ predict precisely the same as the Taniyama-Shimura conjecture.