raising and lowering indices
Once you accept that a vector has two faces — contravariant components v^i with an upper index, and covariant components v_i with a lower index — you need a way to convert between them. Raising and lowering indices is that conversion. It is sometimes nicknamed 'index gymnastics' because in practice it looks like sliding an index up or down at will, but underneath it is a precise geometric operation performed by the metric tensor.
The metric does the lowering: v_i = g_{ij} v^j, summed over j. The inverse metric does the raising: v^i = g^{ij} v_j, where g^{ij} is the matrix inverse of g_{ij} (defined by g^{ik} g_{kj} = delta^i_j). The same works for any tensor and any single index: contract with g to push an index down, contract with g-inverse to push it up, leaving every other index untouched. In flat Cartesian space the metric is the identity, so raising and lowering does nothing and the upper/lower distinction collapses — which is exactly why you never met this operation in introductory vector calculus. In polar, spherical, or curved coordinates the metric is non-trivial, so lowering a component genuinely rescales it (for example, multiplying an angular component by r^2 on the way down).
This operation is the connective tissue of tensor algebra. It lets you take a contraction like the kinetic term g_{ij} v^i v^j and rewrite it compactly as v_i v^i; it converts the gradient (naturally a covector, partial_i phi) into the gradient vector you can follow uphill (g^{ij} partial_j phi); and in relativity it is how you switch between, say, the contravariant four-velocity and its covariant partner. The discipline to keep is that raising then lowering the same index must return you to where you started, and that an index, once summed in an up-down pair, should not be raised or lowered carelessly. A subtle point: in spaces with an indefinite metric (relativity) lowering can flip a sign, so v_0 and v^0 may differ by a minus.
In 2-D polar coordinates the metric is g = [1, 0; 0, r^2] and its inverse is g^{-1} = [1, 0; 0, 1/r^2]. A vector with contravariant components (v^r, v^theta) lowers to covariant components v_r = v^r and v_theta = r^2 v^theta. The angular component is genuinely rescaled by r^2 — proof that lowering is not a do-nothing relabeling once the metric is non-identity.
Lowering an index multiplies by the metric — trivial in Cartesian space, a real rescaling otherwise.
Raising and lowering only relabels which face of the same object you display; it never changes the underlying geometric vector. But it is invertible only where the metric is non-degenerate (its inverse exists) — at a coordinate singularity the operation can break down.