The following bullet points represent the very peak of my understanding of the resolution of the Langlands program for function fields. Disclaimer: I don't know what I'm writing about.
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Drinfeld modules are like the function field analogue of CM elliptic curves. To see this, complexify an elliptic curve $E$ to get a torus $\mathbb{C} / \Lambda$. If $K$ is an imaginary quadratic field, those lattices $\Lambda$ such that $\mathcal{O}_K \Lambda \subseteq \Lambda$ correspond to elliptic curves with CM, meaning that there exists a map $\mathcal{O}_K \to \operatorname{End} E$ whose 'derivative' is the inclusion $\mathcal{O}_K \hookrightarrow \mathbb{C}$. Now pass to function fields. Take $X$ a curve over $\mathbb{F}_q$, with function field $K$, and put $C$ for the algebraic closure of the completion. Then we can define Drinfeld modules as an algebraic structure on a quotient $C / \Lambda$.
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Shtukas are a 'generalisation' of Drinfeld modules. According to Wikipedia, they consist roughly of a vector bundle over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it. From Goss' "What is…" article, I gather that some analogy with differential operators is also involved in their conception.
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Shtukas are used to give a correspondence between automorphic forms on $\operatorname{GL}_n(K)$, with $K$ a function field, and certain representations of absolute Galois groups. For each automorphic form, one somehow considers the $\ell$-adic cohomology of the stack of rank-$n$ shtukas with a certain level structure, and I presume this cohomology has an equivariant structure that gives rise to a representation.
While this gives me a comfortable overview, one thing I cannot put my finger on is why things work the way they work. I fail to get a grasp on the intuition behind a shtuka, and I especially fail to see why it makes sense to study them with an outlook on the Langlands program. This leads to the following questions.
Question 1. What is the intuition behind shtukas? What are they, even roughly speaking? Is there a number field analogue that I might be more comfortable with?
Question 2. How can I 'see' that shtukas should be of aid with the Langlands program? What did Drinfeld see when he started out? Why should I want to take the cohomology of the moduli stack? Were there preceding results pointing in the direction of this approach?
Best Answer
I think shtukas are best understood ahistorically. I would start with the modular curves, but specifically with the (geometric) Eichler-Shimura relation. This says that the Hecke operator at $p$, viewed as a correspondence on $X_0(N) \times X_0(N)$, when reduced at characteristic $p$, is just the graph of Frobenius plus the transpose of the graph of Frobenius. The relevance of this fact to the proof of a Langlands correspondence that relates traces of Frobenius acting on cohomology to eigenvalues of Hecke operators acting on cohomology should be unsurprising, even if completing the argument required much brilliant work by many people.
Now for higher-dimensional Shimura varieties, one cannot always generalize this simple geometric Eichler-Shimura relation, and instead must state and prove a suitable cohomological analogue of it.
When we go to the function field world, on the other hand, it pays to be very naive. We want to define a moduli space of some kind of object where Hecke operators act, and Frobenius acts, and these two actions are related. As David Ben-Zvi described in his answer, we understand what kind of object Hecke operators act on, and how - they act on vector bundles, or more generally $G$-bundles, and they act by modifying the bundle at a particular point in a controlled way. Frobenius also acts on $G$-bundles, by pullback. But these actions have nothing to do with each other.
The solution, then, is to force these actions to have something to do with each other in the simplest possible way - demand that the pullback of a $G$-bundle by Frobenius equal its modification at a particular point, in a certain controlled way. In fact, we can freely do this at more than one point, producing a space on which Frobenius acts like any desired composition of Hecke operators at different points.
The actual way Drinfeld came up with the definition was totally different, and did involve differential operators. He first came up with Drinfeld modules by an inspired analogy with the moduli spaces of elliptic curves. He then realized an analogy with work of Krichever, which he realized should lead to an analogous object defined using sheaves, the resulting definition was a shtuka. Precisely, the relation is that a rank $r$ Drinfeld module is the same thing as a $GL_r$-shtuka with two legs (corresponding to the standard representation of $GL_r$ and its dual under the geometric Satake isomorphism, in the modern language) at two points, where one is allowed to vary and the other is fixed at the point "$\infty$", and furthermore where we require that the induced map of vector bundles at the point $\infty$ is nilpotent.
So moduli spaces of Drinfeld modules will be certain subsets of moduli spaces of shtukas. However, the relationship between Drinfeld modules and shtukas is rarely used to study either - the research on both of them is usually quite separate.