I'm taking a course this semester, and in it we proved that any complete hyperbolic surface is universally covered by $\mathbb{H}^2$. The text, found at http://www.math.brown.edu/~res/Papers/surfacebook.pdf, that we were given for the course, after proving the theorem, discusses only how this result opens the door to many beautiful tilings of the hyperbolic plane.
I asked my professor what the mathematical significance of the result is (I'm still new with the material so I haven't really gotten a hold of the big picture just yet). I can't remember exactly what he said, but it was something along the lines of
"Because of the result, if we wish to study complete hyperbolic surfaces, we can instead study the quotients $\mathbb{H}^2/\Gamma$, where $\Gamma$ is a torsion-free, discrete group of isometries of $\mathbb{H}^2$".
I looked up the Wikipedia article on "Hyperbolic space" http://en.wikipedia.org/wiki/Hyperbolic_space, and in it, it is said (under "Hyperbolic manifolds"),
Every complete, connected, simply-connected manifold of constant negative curvature $−1$ is isometric to the real hyperbolic space $\mathbb{H}^n$. As a result, the universal cover of any closed manifold $M$ of constant negative curvature $−1$, which is to say, a hyperbolic manifold, is $\mathbb{H}^n$. Thus, every such $M$ can be written as $\mathbb{H}^n/\Gamma$ where $\Gamma$ is a torsion-free, discrete group of isometries on $\mathbb{H}^n$.
(I checked Wikipedia to make sure I wasn't remembering incorrectly what my professor told me.)
My question is, how does the fact that the universal cover of any complete hyperbolic surface is $\mathbb{H}^2$ imply that any suchcomplete hyperbolic surface can be written $\mathbb{H}^2/\Gamma$?