This is just a rephrasing of Deane's answer, but let me add one general comment. To any Riemannian metric (or pseudo-Riemannian metric) $g$ on a manifold $M$, you can associate a Levi-Civita connection $\nabla : T_M \to T_M \otimes \Omega_M$, where $T_M$ is the cotangent sheaf and $\Omega_M$ is the cotangent sheaf. Just like how the de Rham $d : \mathcal{O}_M \to \Omega_M$ can be extended to $d : \Omega_M^i \to \Omega_M^{i+1}$, the connection $\nabla : T_M \to T_M \otimes \Omega_M$ can be extended to $\nabla : T_M \otimes \Omega_M^i \to T_m \otimes \Omega_M^{i+1}$.

Then being able to find local coordinates $x_i$ such that $g_{ij} = \delta_{ij}$ is equivalent to $\nabla^2 = \nabla \circ \nabla : T_M \to T_M \otimes \Omega_M^2$, which corresponds to the Riemann curvature tensor that Deane mentions, being zero. This should be reminiscent of the $d^2 = 0$ of de Rham cohomology, or homological algebra in general... ;-)

It is perhaps not-so-standard (or not as standard as I would like) to talk about connections in terms of sheaves, but this point of view is better because it generalizes better. What I said above works in all of the "standard" geometric categories: $C^\infty$, real analytic, complex analytic, super-, etc. (Maybe it even works in the algebraic category? But I'm not sure, as the notion of local coordinates is more complicated in algebraic geometry.)