Fix $n$ and a field $k$ of characteristic zero. Let $G$ be the proalgebraic group of automorphims of $k[[x_1,...x_n]]$. Let $G_0$ be the subgroup of automorphisms preserving the closed point (note that for general $T$, $G_0(T)$ can be a proper subgroup of $G(T)$). Let $X$ be a regular variety over $k$ and let $P$ be the principal $G$ bundle of formal coordinate systems, naturally a $G$ torsor over $X$. I hear that there is a connection between $P$ and $D_X$modules. what is this connection?

Assume X is ndimensional and regular. Then there is a functor from Gmodules V to D_{X}modules, given by an associated bundle construction. Take the trivial (ind)bundle on P with fiber V, and quotient by the action of G on P and V. If you replace G with G_{0} and P with the canonical G_{0}torsor, the same construction yields an O_{X}module. The extra structure of a Gaction lets you identify infinitesimally nearby fibers. 

