There are of course lots of definitions and references for this, but in the same way that, on a manifold $M$, a Riemannian metric is a section of positive definite symmetric bilinear forms on $TM$ ...
A spinor which belogs to a representation of a group $G=SO(p,q)$ is a section of a product bundle $S(M)\otimes E$, where $S(M)$ is a spin bundle over a four dimensional orientable and compact manifold ...
Let us consider an orientable smooth 4-manifold $M$. Pick a vector bundle $T$ that's isomorphic to the tangent bundle $TM$. We then equip $T$ with a cotetrad (or coframe field) $e$ and (spin) ...