A Killing spinor on a (pseudo-)Riemannian manifold is a covariantly constant spinor : a section of some spinor bundle that is annihilated by the covariant derivative of the corresponding Levi-Civita connection.
A discussion with an eye towards applications in supersymmetry is around page 907 in volume II of
Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)