Definition:Principal Ultrafilter/Nonprincipal

Definition
Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ denote the power set of $S$.

Let $\mathcal F \subset \mathcal P \left({S}\right)$ be an ultrafilter on $S$ which does not have a cluster point.

Then $\mathcal F$ is a nonprincipal ultrafilter on $S$.

Also known as
A nonprincipal ultrafilter is also known as a free ultrafilter.

Also see

 * From Cluster Point of Ultrafilter is Unique, an ultrafilter is either principal or nonprincipal -- there are no other options.