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.