the cut locus
Walk away from a fixed home point p in every direction along geodesics, always taking the shortest route. For a while each geodesic really is the unique shortest path home. But sooner or later, in each direction, you reach a point past which that road is no longer the shortest way back — either a rival geodesic now ties it, or the road simply stops being a minimizer. The collection of all these 'last moments of shortest-ness,' one along each direction, is the cut locus of p. It is the precise edge of the region where geometry is simple.
Precisely: along a unit-speed geodesic gamma from p, the cut point is gamma(t_c) where t_c is the largest time such that gamma minimizes distance from p on [0, t_c]. At the cut point exactly one of two things happens: either gamma(t_c) is the first conjugate point of p along gamma, or there is a second, distinct minimizing geodesic from p to gamma(t_c) (a tie). The cut locus Cut(p) is the union of all cut points over all directions. Its complement is a star-shaped open set on which exp_p is a diffeomorphism, so M = (a metric ball image) glued along Cut(p): the whole manifold is the exponential image of a domain in T_p M, with all the identifications happening on the cut locus. The distance function d(p, .) is smooth exactly off Cut(p) and the point p itself.
The cut locus controls global shape and topology. The injectivity radius at p is the distance to the nearest cut point, and a lower bound on it is what lets local estimates patch into global ones; the diameter is bounded by curvature plus cut-locus control (Bonnet-Myers). On a flat torus the cut locus of a point is the 'far' grid of edges where shortest paths split; on a sphere the cut locus of the north pole is the single antipodal south pole. Honest subtlety: the cut locus can be complicated — generically a stratified set, possibly not a smooth submanifold — and computing it explicitly is hard even for innocent-looking metrics; on an ellipsoid it is already a nontrivial arc, not a point.
On the round sphere S^2, the cut locus of the north pole is a single point — the south pole — which is also its first conjugate point (the 'conjugate' case). On a flat square torus R^2 / Z^2, the cut locus of the origin is the image of the boundary of the unit square centred at the origin: a one-dimensional graph (a figure made of edges and a vertex) where two or more shortest paths tie, with NO conjugate points involved (flat space has none) — pure 'tie' case. The two examples show the two distinct mechanisms by which minimizing ends.
Sphere: cut locus = conjugate antipode (one point). Flat torus: cut locus = a graph of tie points, no conjugate points.
The cut point comes at or before the first conjugate point, never after. So 'a geodesic minimizes up to its cut point' is the sharp statement; 'minimizes up to its first conjugate point' is only an upper bound on how far it could possibly minimize.