My own "go-to" introduction is Euler characteristic. What is nice about it is that you can have tons of audience participation.
First draw a bunch of polyhedra on the board (or have models that you can distribute to the audience). Ask people to count the number of faces, edges and vertices. Write a few of these down on the board. Ask if anyone sees any patterns. Usually at least one person will notice that F+V-E=2.
A combinatorial proof of this, by first reducing to the planar case, triangulating, and then removing triangles from the outside in, showing that at each stage you are leaving F+V-E invariant, is something I have had success with even with highschool students (at least one on one). At each stage you can have someone in the audience confirm the invariance ("What happens to F+V-E when I take a triangle like this away?") Have a triangulated torus already prepared. Observe that F+V-E=0. Tell them that in general an "n-holed" donut has F+V-E=2-2g where g is the number of holes. So somehow this number F+V-E depends on the whether you could stretch one of these shapes into the other, but not on the rigid geometry. Explain that a similar combinatorial proof would be difficult for the n-holed donut, but their is an entire subject called "algebraic topology" which has developed machinery which makes this kind of result easy to see.