To add to what Ben said: Typically the resolution of the sheaf mapping ${\frak{g}}\to\Gamma(TX)$ is the Spencer resolution, i.e., a canonical $G$-equivariant exact sequence of differential operators $D_i:\Gamma(V_i)\to \Gamma(V_{i+1})$ for $0\le i\le n$ where $V_0 = TX$ and $\mathrm{ker}(D_0) = {\frak{g}}$. When you have a flat $(G,X)$-geometry on a manifold $M$, this sequence pulls back (under the local charts) to define the corresponding sequence on $M$.
For example, when $G$ is the group of isometries of a $3$-dimensional Riemannian manifold of constant curvature, one finds that the ranks $r_i$ of the bundles $V_i$ are $$ (r_0,r_1,r_2,r_3) = (3,6,6,3) $$ and the order of the differential operators $D_0$ and $D_2$ is $1$ while the order of the differential operator $D_1$ is 2. If $G$ is the group of conformal transformations of flat $3$-space, then the ranks are $$ (r_0,r_1,r_2,r_3) = (3,5,5,3) $$ and the order of the differential operators $D_0$ and $D_2$ is $1$ while the order of the differential operator $D_1$ is 3.
Look in any book that treats the Spencer resolution or the topic of Lie equations for references. (I'm traveling, so I don't have access to the literature.)