A Killing spinor on a (pseudo-)Riemannian manifold is a spinor – a section of some spinor bundle $v \in \Gamma(S)$ – that is taken by the covariant derivative of the corresponding Levi-Civita connection to a multiple of itself
for some constant $\kappa$.
If that constant is 0, hence if the spinor is covariant constant, then one also speaks of a covariant constant spinor or parallel spinor (with respect to the given metric structure).
Similarly a Killing vector is a covariantly constant vector field.
Pairing two covariant constant spinors to a vector yields a Killing vector.
In supergravity, super spacetimes which solves the equations of motion and admit Killing spinors are BPS states (at least if they are asymptotically flat and of finite mass).
Lecture notes include
A discussion with an eye towards applications in supersymmetry is around page 907 in volume II of
Discussion relating to special holonomy includes
Discussion relating to Killing vectors in supergeometry includes
Original articles include
Thomas Friedrich, Zur Existenz paralleler Spinorfelder über Riemannschen Mannigfaltigkeiten Czechoslavakian-GDR-Polish scientific school on differential geometry Boszkowo/ Poland 1978, Sci. Comm., Part 1,2; 104-124 (1979)
Thomas Friedrich, Zur Existenz paralleler Spinorfelder über Riemannschen Mannigfaltigkeiten, Colloquium Mathematicum vol. XLIV, Fasc. 2 (1981), 277-290.
Discussion relating to G2-structures includes
Discussion in 11-dimensional supergravity includes