Nineteen proofs of Euler's Formula, from The Geometry Junkyard. (This is the one that says that if V, E, and F are the number of vertices, edges, and faces of a polyhedron, then V-E+F=2.)
This is something of an embarrassment of riches; I want one for my class tomorrow, because we're talking about regular polyhedra, and to talk about that without mentioning Euler's formula would be remiss.
I'll probably give "Proof #8: Sum of angles" from the page, which sums up the various angles involved in a drawing of the graph corresponding to a polyhedron, or the proof that Wikipedia attributes to Cauchy, which triangulates that graph (not changing V-E+F) and then removes edges one or two at a time (still not changing V-E+F). These seem the most accessible. (They're also the ones I managed to struggle in the general direction of while walking to the coffee house a couple hours ago.)
If I wanted to scare the hell out of my students (this class has no prerequisites), I'd give the "binary homology" proof.