the covariant derivative
Try to differentiate a vector field in curved space and you hit a snag: to compare a vector here with a vector a step away, you must subtract them, but they live in different tangent spaces oriented differently. The plain partial derivative mixes up the real change of the vector with the mere turning of the coordinate grid, and the result is not a proper geometric object. The covariant derivative repairs this, giving a derivative that all observers agree on regardless of their coordinates.
It adds a correction, built from the Christoffel symbols, that cancels the spurious coordinate-turning: for a contravariant vector, nabla_a V^b = partial_a V^b + Gamma^b_ac V^c, while for a covariant one the correction flips sign, nabla_a W_b = partial_a W_b - Gamma^c_ab W_c. The outcome is a genuine tensor, one rank higher than what you started with. A defining property in general relativity is metric compatibility, nabla_a g_bc = 0, meaning the operation of measuring lengths commutes with differentiation, which is what fixes the connection to be the unique Levi-Civita one.
The covariant derivative is the tool that lets you write physical laws in a form valid in any spacetime: wherever a flat-space law contains an ordinary derivative, you promote it to a covariant derivative (the 'minimal coupling' rule), and the law becomes generally covariant. The single most consequential fact about it is that covariant derivatives do not commute. The failure of nabla_a nabla_b to equal nabla_b nabla_a is not sloppiness; that commutator is precisely the Riemann curvature tensor, so non-commuting derivatives ARE curvature.
For a vector held at constant components in polar coordinates, the plain partial derivative reports zero change, yet the covariant derivative may be nonzero: as you move, the radial and angular basis directions themselves swing round, and the vector really is turning relative to a fixed frame.
The covariant derivative sees the real change the partial derivative misses.
Promoting partial to covariant derivatives is unambiguous only for first derivatives; because covariant derivatives fail to commute, a second-derivative term can be ordered in inequivalent ways, and the difference is a curvature term. Minimal coupling is a rule of thumb, not a theorem.