a geodesic metric space
Imagine standing at one point and wanting to walk to another by the shortest possible route, and being able to actually find such a route every time. A geodesic metric space is a metric space in which any two points can in fact be joined by a shortest path — a curve whose length equals the distance between them. It is a length space where the infimum is not just approached but achieved.
Make the path notion precise: a geodesic from x to y is an isometric embedding gamma: [0, d(x,y)] -> X with gamma(0) = x, gamma(d(x,y)) = y, meaning d(gamma(s), gamma(t)) = |s - t| for all s, t. Such a curve is parametrized by arc length and is everywhere a shortest path between its own endpoints; in this metric-geometry usage 'geodesic' means globally shortest, which is stronger than the Riemannian usage where a geodesic only need be locally straight. A space (X, d) is a geodesic metric space when such a gamma exists for every pair x, y. Note geodesics may fail to be unique — between the north and south poles of a sphere there are infinitely many.
This is the workhorse setting for curvature comparison: the definitions of CAT(k) and Alexandrov curvature bounds all compare the triangles you build out of geodesics, so you first need geodesics to exist. The Hopf-Rinow theorem for length spaces says a complete, locally compact length space is automatically geodesic, which is why so many natural examples — complete Riemannian manifolds, the Cayley graph of a group, finite-dimensional CAT(0) spaces — qualify. Beware the standard trap: completeness alone is not enough without local compactness, and a length space can be complete yet have a pair of points joined by no shortest path.
Any complete Riemannian manifold, by Hopf-Rinow, is a geodesic metric space: between any two points there is a length-minimizing geodesic. A finite connected graph, with each edge given length 1 and distance measured by shortest paths, is also geodesic — a shortest walk between vertices realizes the distance, and we travel along edges at unit speed. By contrast the open unit disc in R^2 (without its boundary), with the intrinsic length metric, is a length space but not geodesic relative to certain boundary-approaching pairs, and removing a single interior point breaks geodesicity outright.
Geodesic spaces: complete manifolds and graphs realize distances by actual shortest paths; an open or punctured disc may not.
Do not assume geodesics are unique. Uniqueness is a special, strong property — it holds in CAT(0) spaces but fails on the sphere and on a flat cylinder, where multiple shortest paths can connect the same two points.