$$Bun_n = \{\text{set of isomorphism classes of $n$ rank (holomorphic) vector bundles on curve } X \}$$

$Bun_n$ is a moduli space and an algebraic stack, so a moduli stack! The set would be countable, as there are finitely many vector bundles over each finite field extension. One is then interested in studying a restricted (cuspoidal) space of functions on $Bun_n$.

Well, the Hecke operators (which correspond to modifications at points) from classical Langlands can be geometrized as well. They can be projected to the $Bun_n$ as well. We can define Hecke eigensheaves on $Bun_n$ which correspond to moduli of the rank $n$ local systems on the curve $X$. This is the geometric Langlands correspondence. There is the Hecke correspondence between the Hecke stack and the moduli stack.

Moreover, these Hecke eigensheaves are D-modules on $Bun_n$ satisfying a certain property set by the vector bundle $E$.

Now, a very short remark on why $Bun_n$ is an algebraic stack but not a scheme. Because the moduli space has non-trivial automorphisms. Moreover, a general scheme can not track the automorphisms, so if you were to construct a scheme $Bun_n$, you would fail to account for the automorphism groups. Moreover, the stack structure on $Bun_n$ is important for Langlands correspondence as well, even for defining D-modules on it or for the Hecke correspondence to work. We will look into the Hecke eigenvsheaves and what comes before all of these geometrizations in some later post.