normal coordinates
If you want curved space to look as flat as possible near a chosen point — so that, right there, your formulas reduce to ordinary Euclidean ones — you use normal coordinates. They are the coordinate system in which geodesics through the chosen point become straight lines through the origin, and the metric agrees with the Euclidean metric to first order at that point. They are the geometer's standard way to 'work at a point.'
To build them: fix p in M, choose an orthonormal basis e_1, ..., e_n of T_p M, and use the exponential map. A point near p gets the coordinates (a^1, ..., a^n) where exp_p(a^1 e_1 + ... + a^n e_n) is that point — i.e. you read off a tangent vector's components and push it out along its geodesic. In these coordinates two clean facts hold AT p (and only exactly at p): the metric is the identity, g_ij(p) = delta_ij, and all Christoffel symbols vanish, Gamma^k_ij(p) = 0. Equivalently the metric's first derivatives vanish at p, so the metric is Euclidean up to second order; the second-order correction is precisely the curvature, g_ij = delta_ij - (1/3) R_ikjl a^k a^l + ... .
Why they are indispensable: many tensor identities are easiest to prove by choosing normal coordinates at a point, where the messy Christoffel terms drop out, computing, and then noting the result is tensorial hence coordinate-free. The honest caveat is in the phrase 'at p': you can kill g's first derivatives at ONE point, but you cannot kill the SECOND derivatives unless curvature actually vanishes. That residual second-order term is the coordinate-free obstruction to flatness — curvature is exactly what normal coordinates cannot iron out.
On the sphere, normal coordinates centered at the north pole are essentially (up to scaling) the azimuthal-equidistant map projection used on the UN flag: geodesics (meridians) are straight rays from the center, and distances from the pole are read off radially. The distortion that grows as you move outward is the visible footprint of curvature, the term normal coordinates cannot remove.
Azimuthal-equidistant projection is the sphere's normal-coordinate chart at the pole; the outward distortion is curvature.
Normal coordinates flatten the metric to FIRST order at a single point only (g_ij = delta_ij, Gamma^k_ij = 0 there). The second-order term is curvature and cannot be removed by any coordinate change — that is the precise sense in which curvature is intrinsic.