Elevator pitch for a (sub)field of maths?
Combinatorics: Counting without your fingers
There's the old "Topology: Geometry on a rubber sheet."
For both Category Theory and Universal Algebra, I also like The Esperanto of Mathematics/Algebra (except for the negative connotation of "nobody actually uses it"...).
Algebra: The mathematics of structures.
Set Theory: The main foundation of mathematics.
Logic: The main foundation of set theory.