The central idea behind the Geometric Langlands Correspondence (GLC) is that arithmetic, geometry, and symmetry are different manifestations rather than distinct objects. GLC is a statement about the nonabelian analog of the Fourier tranform. The Fourier transform concerns the spectral data of certain 'functions'. In GLC, we study the moduli spaces of bundles and local systems rather than linear spaces, and we replace functions by geometric objects such as sheaves.

Given a smooth projective algebraic curve $X$, there is a moduli stack Bun$_G$ on $X$ of principal $G$-bundles. (In the Quill 16, we attempted to describe the Bun$_G$.) This moduli stack parameterizes the principal $G$-bundles on $X$. We know that $X$ is a compact Riemann surface (because every compact Riemann surface is a projective variety), and if it has no punctures, marked points, or modifications of divisors, then it is called the unramified case. So, in such a case, the simplest thing is to assume there are no singular points or anything exceptional happening at a point. However, in the case of number fields, ramification is inescapable. The case of ramified (and local investigations of the ramification points) is still being largely studied in GLC. Anyway, the modern statement of GLC is an equivalence of (derived categories of) differential equations (called $\mathcal{D}-$modules) on one side and ind-coherent sheaves (with nilpotent singular support) on the other side (also called the spectral side) which is described using the Langlands dual group $^LG$: $$\mathcal{D}\text{Bun}_G \sim \text{IndCoh}_{\mathcal{N}} (\text{Loc}_{^LG})$$

where $\text{Loc}_{^LG}$ is the moduli stakc of $^LG$-local systems on $X$. Classically, the spectral parameters are Galois representations, but geometrically, they are defined using local systems. One can think of them as a flat connection or a representation. Now, a category of quasi-coherent sheaves is not sufficient on (a derived stack) $\text{Loc}_{^LG}$ because it cannot see the singular geometry, for which Gaitsgory (and Rosenbluym) introduced Ind-Coherent sheaves $\text{IndCoh}$ and $\mathcal{N}$ represents a nilpotent cone (see here?).  Roughly speaking, GLC predicts the geometric objects in the moduli stack of $G$-bundles by studying the spaces for the local systems on $^LG$.

In physics, geometric Langlands emerges as S-duality in supersymmetric gauge theory in four dimensions. A ramified case has also been worked on by Gukov and Witten in this paper. While the subject continues to thrive and I continue to work on my poor understanding of it, it is a far-reaching web of mathematics and physics that is being developed at a remarkable rate by remarkable standards.