The geodesics in a complete Riemannian metric go on indefinitely, i.e., each geodesic is isometric to the real line. For example, Euclidean space is complete, but the open unit disk is not complete since any geodesic ends at a finite distance. Whether or not a manifold is complete depends on the metric.
For instance, the punctured plane 
is not complete with the usual metric. However, with
the Riemannian Metric 
, the punctured
plane is the infinite (flat) cylinder, which is complete. The figure above illustrates
a geodesic which can only go a finite distance because
it reaches a hole in the punctured
plane, exemplifying that the punctured plane with the usual metric is not complete.
The path 
is a geodesic parametrized by arc length.
Any metric on a compact manifold is complete. Consequently, the pullback metric on the universal cover of a compact manifold is complete.
