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.
Similarly a Killing vector is a covariantly constant vector field.
A discussion with an eye towards applications in supersymmetry is around page 907 in volume II of
Pierre Deligne, P. Etingof, Dan Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and Edward Witten, eds.
Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)