One of the things I like to mention, since I study topology, is the Brouwer fixed point theorem. The idea to explain is that if you pick up a piece of paper, DON'T RIP IT, but crumple it, turn it over, fold it, whatever, put it down on top of another one, then there will always be at least one point that will match up with the one below it on the other paper. It's very physical, very counterintuitive, and thoroughly math, though it's better demonstrated with a decorated napkin than plain.
Alternatively, in the same vein, one can talk about the hairy-sphere theorem (the idea that you can't comb a hairy sphere all the way around without a cowlick; i.e., you can't have a nonvanishing continuous vector field along the sphere).
