parallel transport
You are holding a spear and you walk along a path on a curved surface, taking care never to deliberately turn the spear — at each tiny step you keep it 'as parallel as possible' to its previous direction. Parallel transport is the precise rule for doing this: carrying a tangent vector along a curve while keeping it from rotating relative to the space itself. On a flat plane this just slides the arrow without turning it; on a curved surface something surprising happens, and that surprise is curvature made visible.
Precisely, given the Levi-Civita connection, a vector field V is parallel-transported along a curve when its covariant derivative along the curve is zero: nabla_(dot c) V = 0, where dot c is the curve's velocity. In coordinates this is a system of ordinary differential equations driven by the Christoffel symbols, with a unique solution once you fix the starting vector — so parallel transport gives a definite, length-and-angle-preserving way to carry any vector from the start of a curve to its end. It is exactly the construction that lets you compare vectors at different points, the comparison that bare tangent spaces lacked.
The decisive discovery is that the result depends on the path, not just the endpoints. Carry a vector around a closed loop on a curved surface and it generally comes back rotated by a definite angle; the amount of rotation around small loops is precisely the curvature, and the rotation around large loops (the holonomy) encodes the global geometry. This is the cleanest hands-on definition of curvature: a sphere is curved because a vector marched around a triangle returns turned, while on the flat plane it always returns unchanged. Geodesics can now be re-described beautifully as the curves that parallel-transport their own velocity.
On a sphere, start at the north pole pointing a vector south along one meridian. Carry it down to the equator (keeping it tangent and not turning), then along the equator a quarter of the way around, then back up to the north pole. The vector returns to the pole rotated by 90 degrees, even though you never turned it — the rotation equals the enclosed area times the sphere's curvature.
Marched around a closed loop and brought home rotated — that rotation is curvature you can feel directly.
Parallel transport is path-dependent on a curved space: the same vector carried by two different routes to the same endpoint usually disagrees. Independence of path is equivalent to flatness, so this dependence is not a defect but the very signature of curvature.