I understand this statement of Uniformization Theorem:
Every simply connected Riemann surface is conformally equivalent to the unit disk, the complex plane, or the Rieman $n$ sphere.
However, the following statement totally throws me out:
Every connected 2-manifold is diffeomorphic to a quotient of one of the three constant curvature model surfaces listed above by a discrete group of isometries acting freely and properly discontinuously. Therefore, every connected 2-manifold has a complete Riemannian metric with constant Gaussian curvature.
From the context, "the three constant curvature model surfaces listed above" are zero curvature, $1/R^2$ and $-1/R^2$.
Could anyone explain how these two are equivalent? I am particularly lost at the terms "a quotient of" and "a discrete group of isometries acting freely and properly discontinuously".