A Killing spinor on a (pseudo-)Riemannian manifold is a spinor – a section of some spinor bundle that – 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).
Pairing two covariant constant spinors to a vector yields a Killing vector.
Lecture notes include
A discussion with an eye towards applications in supersymmetry is around page 907 in volume II of