A Killing spinor on a (pseudo-)Riemannian manifold is a spinor – a section of some spinor bundle – that is taken by the covariant derivative of the corresponding Levi-Civita connection to a multiple of itself
for some constant .
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).
Notice however that a Killing spinor with non-vanishing may be understood as a genuine covariantly constant spinor, but with respect to a super-Cartan geometry modeled not on super-Euclidean space/super-Minkowski spacetime, but on its spherical/hyperbolic or deSitter/anti-deSitter versions spring
Pairing two covariant constant spinors to a vector yields a Killing vector.
Lecture notes include
Discussion regarding the conceptualization of Killing spinors in super-Cartan geometry is in
Discussion relating to special holonomy includes
A discussion with an eye towards applications in supersymmetry is around page 907 in volume II of
Discussion of classification includes
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