One proof not yet cited is by Ricci flow. It is a proof of the differential geometric version of Uniformization : in each conformal class of Riemannian metric, there is a metric of constant curvature. The idea is very natural : by construction, metric of constant curvature are fixed points of Ricci flow so take a general metric, evolve it by Ricci flow and show that it converges. This is essentially a work of Hamilton (reference : "The Ricci flow :an introdution" Chow, Knopf) which was completed by Chen, Lu and Tian.
I don't claim that it is the best proof of unifomization theorem, it is just a way to see in a simple case how the Ricci flow can be used to obtain classification results. (The point is that for surfaces Ricci flow has no singularity in finite time, which is not the case in 3 dimensions).