\mathcal{D}-modules are very important objects in algebraic geometry and mathematical physics (for example, in string theory). In a previous post, we saw that Hecke eigensheaves on a Bun_G (a set of isomorphisms of vector bundles) are \mathcal{D}-modules. They are a quasi-coherent sheaf of the scheme of differential operators.
As the name suggests, \mathcal{D}-modules are just modules over the ring of differential operators on some variety X.
Let X be a smooth variety over a field k (of char 0), then O_X is the structure sheaf over it. Let \mathcal{F} be a quasi-coherent sheaf over O_X. Let us denote D as a differential operator and for open affine subset U\in X, \mathcal{D}(U) denotes the ring of differential operators on X. Then U \to D(U) is a quasi-coherent sheaf of O_X-modules. And we call this sheaf of differentials operators \mathcal{D}-modules.
\mathcal{D} is sheaf of non-commutative algebra. A very good example is Weyl Algebra A_n(k). Locally, they are generated by the Heisenberg commutation relations here. Globally, we are interested in seeing if a solution exists by gluing them. So, \mathcal{D}-modules help us to understand local-global pictures of linear differential equations.
There is a specific class of D-modules that were of interest to Kashiwara - holonomic modules in the Riemann-Hilbert correspondence. Anyway, the central concept to understand in \mathcal{D} is that they are a sheaf of modules over the ring of differential operators and they are quasi-coherent to O_X-modules. For physicists, geometric Langlands has been putting the duality between the category of D-modules on Bun_G and the Fukaya category of Lagrangians in its cotangent bundle. Moreover, \mathcal{D}-modules give a geometric meaning to the automorphic forms.
