a minimizing geodesic
Stretch a string tight between two points on a curved surface and it traces the shortest path that stays on the surface. On a sphere that is an arc of a great circle; on a flat plane it is a straight segment. A minimizing geodesic is exactly that: among all curves on a Riemannian manifold M joining two given points, one whose length is as small as possible. It answers the most basic question of geometry — what is the shortest route from p to q?
Two ideas have to be kept apart. A geodesic is a curve that is locally straight: it solves the geodesic equation nabla_{gamma'} gamma' = 0, meaning it has zero acceleration as felt inside the manifold, so it never turns. A minimizing geodesic is the stronger, global notion: its total length really is the infimum over all competitor curves. Every minimizer is a geodesic (a shortest curve cannot afford to wiggle), but the converse fails — a geodesic is only guaranteed to minimize for short enough times. Concretely, on the unit sphere the equator is a geodesic, yet going three-quarters of the way around it is far from shortest; the short arc the other way wins. The length L(gamma) = integral over [a,b] of |gamma'(t)| dt is what we minimize, and any minimizer can be reparametrized to have constant speed.
Minimizing geodesics are the bridge from the metric (distance function d) to the differential structure (curves and the connection). The distance d(p,q) is the infimum of lengths; a minimizing geodesic is a curve that achieves it. Whether one exists at all is not automatic — on the punctured plane the two sides of the missing origin may have no shortest connector — and the Hopf-Rinow theorem is precisely the statement that completeness guarantees existence. Where a geodesic stops minimizing is governed by conjugate points and the cut locus, the central characters of comparison geometry.
On the round sphere S^2 of radius 1, take p the north pole and q a point at colatitude theta (with 0 < theta < pi). The meridian arc from p to q has length theta and is the unique minimizing geodesic. The complementary arc of the same great circle, going the long way, has length 2pi - theta; it is still a geodesic but minimizes nothing. If q is the south pole (theta = pi) every meridian is a minimizer — minimizers need not be unique.
Geodesic versus minimizing: both sphere arcs are geodesics, only the short one minimizes; antipodal points show non-uniqueness.
A common error is to say 'geodesic = shortest path.' Geodesics only minimize locally; past a conjugate point or the cut locus they no longer minimize, even though they remain perfectly good geodesics.