The DeRham space is a stack $X_{DR}$ associated to a smooth variety $X$, so that modules on $X_{DR}$ are D-modules on $X$. This is accomplished by declaring the maps from $Y$ into $X_{DR}$ are the same as maps from $Y^{red}$ (the reduced scheme) into $X$. This has the effect of identifying points with their infinitesmal neighborhoods.
The DeRham space is often most useful as a conceptual tool. However, a specific application of it was by Ben-Zvi and Nevins, who used it (and other tools) to show that certain cusped versions $\widetilde{X}$ of $X$ had equivalent categories of D-modules. The idea being, these cusps were identifying some of the infinitesmal neighborhoods of some of the points, and so they should be intermediate between a variety and its DeRham space.