(differential geometry:)
A path which is locally shortest. That is, for any point on the path, all points on the path within a certain distance are connected along the path by the shortest possible route.

On Earth, geodesics are given by great circles, i.e. circles in a plane going through the centre of the Earth. Such paths are obviously not necessarily shortest (consider going from London to Paris by flying over the Atlantic, and coming back through Asia and Europe). But for "sufficiently close" points (anything in the same hemisphere), great circles are shortest paths. This is why airlines fly great circles.

On a smooth manifold, this metric characterisation of a geodesic leads to a differential geometric characterisation.