By no means trivial, a simple characterization of a mathematical structure is a simply-stated one-liner in the following sense:
Some general structure is (surprisingly and substantially) more structured if and only if the former satisfies some (surprisingly and superficially weak) extra assumption.
For example, here are four simple characterizations in algebra:
- A quasigroup is a group if and only if it is associative.
- A ring is an integral domain if and only if its spectrum is reduced and irreducible.
- A ring is a field if and only its ideals are $(0)$ and itself.
- A domain is a finite field if and only if it is finite.
I'm convinced that there are many beautiful simple characterizations in virtually all areas of mathematics, and I'm quite puzzled why they aren't utilized more frequently. What are some simple characterizations that you've learned in your mathematical studies?