The differential geometry of manifolds with spin structure is called spin geometry. It studies spin group-principal bundles, spin representations, the associated spinor bundles, and the Dirac operators acting on spaces of sections of these bundles, hence also their index theory and generally K-theory.
The relevance of spin geometry in physics rests on the fact that in quantum mechanics and quantum field theory in general and in the standard model of particle physics in particular, fermions such as the electron are mathematically modeled as sections of spin-bundles. The very term spin originates in the fact that the quanta of these fields behave to some extent as if they had an intrinsic angular momentum, as if they were spinning about an axis as a classical top.
Spin geometry also plays a central role in supersymmetric quantum field theory such as supergravity.
The classical monograph on spin geometry is
Fundamentals of the relevant supergeometry are in
in P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison, E. Witten (eds.) Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)
Discussion in the context of conformal geometry is in