The universal covering space is a topological construction that makes no reference to smooth or Riemannian structures. If the base is a smooth manifold, then the universal covering space has a natural smooth structure defined by pulling back the smooth structure on the base by the universal covering map, and by construction the universal covering map becomes a local diffeomorphism.
If the base is a Riemannian manifold, then the pullback of the metric by the universal covering map is again a Riemannian metric, making the universal covering a simply connected Riemannian manifold, the universal covering map a local isometry, and the deck group acts by isometries.
If the base has a Riemannian metric with constant sectional curvature, then the universal covering space is isometric to one of the constant-curvature space forms and the base can be realized as the quotient of a space form by a representation of its fundamental group in the isometry group of the space form. This is very useful in low dimensions for translating topological questions into a form amenable to analysis by geometric tools.