A Killing vector field is an infinitesimal isometry.
For $(X,g)$ a Riemannian manifold (or pseudo-Riemannian manifold) a vector field $v \in \Gamma(T X)$ is called a Killing vector field if it generates isometries of the metric $g$. More precisely, if, equivalently
the Lie derivative of $g$ along $v$ vanishes: $\mathcal{L}_v g = 0$;
the flow $\exp(v) : X \times \mathbb{R} \to X$ is a flow by isometries.