a conjugate point
Stand at the north pole and shoot geodesics out in every direction. They all spread apart at first — but on a sphere they all refocus and meet again at the south pole. That meeting place, where infinitely many geodesics from one point reconverge, is the conjugate point of the start. It is the moment a curved space 'catches up' with the spreading rays and pulls them back together, and it marks exactly where a geodesic stops being the shortest route.
Precisely: given a geodesic gamma starting at p = gamma(0), a point q = gamma(t*) with t* > 0 is conjugate to p along gamma if there is a nontrivial Jacobi field J with J(0) = 0 and J(t*) = 0. By the variation picture, this means there is a one-parameter family of geodesics all leaving p that, to first order, all arrive back at q — neighbouring geodesics from p focus there. Equivalently, the differential of the exponential map d(exp_p) is singular at the vector t* gamma'(0): exp_p fails to be a local diffeomorphism there, so geodesic 'rays' from p pile up. The number of independent vanishing Jacobi fields is the multiplicity (1 for a sphere's antipode in dimension 2, n-1 for the round S^n).
Conjugate points are the local obstruction to minimizing. The key theorem: a geodesic gamma does NOT minimize length past its first conjugate point — strictly beyond t*, you can always find a shorter nearby curve. This is the Jacobi criterion, proved via the second variation of energy: before the first conjugate point the index form is positive definite (the geodesic is a strict local minimum of energy); at the first conjugate point it becomes degenerate; after it, negative directions appear. The Morse index theorem makes this quantitative by counting conjugate points. Honest caveat: failing to minimize past a conjugate point is a one-way statement — a geodesic can also stop minimizing earlier, at the cut locus, before any conjugate point is reached, when a different geodesic from p ties it in length.
On the round sphere S^n of radius 1 the south pole is conjugate to the north pole along every meridian, at distance pi, with multiplicity n-1 (all the Jacobi fields |J| = sin t vanish there at once). On flat R^n there are NO conjugate points at all — the Jacobi equation is J'' = 0, so a field vanishing at 0 grows linearly and never returns to zero — which is the analytic shadow of the fact that straight lines in R^n minimize forever. On a hyperbolic space (K = -1), |J| = sinh t likewise never returns: negative curvature kills conjugate points entirely.
Positive curvature creates conjugate points (sphere: at distance pi); zero and negative curvature have none.
Do not confuse conjugate points with cut points. A geodesic stops minimizing AT the cut point, which comes at or before the first conjugate point — never after. Past a conjugate point it certainly fails to minimize, but it may have failed earlier.