There is this nice quote whose wording I can't quite recall. It is something like "physics is the study of the laws of God. Mathematics is the study of the laws even God must follow."
I think there are some nice elementary examples of this in certain areas of combinatorics. Consider, for example, Ramsey numbers: if $6$ people are at a party, either $3$ of them all know each other or $3$ of them all don't know each other (but this is not true for $5$ or less people). That's something most people don't know, it's really easy to demonstrate by picking six people from the audience, and it is of a totally different flavor from the "equational" mathematics most people are familiar with. (Caveat: I have never actually tried this demonstration.) You can then continue: if $18$ people are at a party, then either $4$ of them all know each other or $4$ of them all don't know each other (but this is not true for $17$ or less people).
Then you continue: the corresponding best number for $5$ people is not known. This is just about the easiest most easily stated open problem I knowhow to state, and it is a good way to show people that mathematics is not "finished" in any meaningful sense. If you were sufficiently handwavy and included lots of pictures, it might even be possible for you to sketch the proof that all the Ramsey numbers exist.
Another potentially good example is Hall's marriage theorem, especially if you use the marriage-theoretic terminology the entire time. I saw a lecturer do this recently and it was quite funny. Again, if you use enough pictures, this might be manageable to sketch.