Definition:Principal Ultrafilter/Nonprincipal

Definition
Let $S$ be a set.

Let $\powerset S$ denote the power set of $S$.

Let $\FF \subset \powerset S$ be an ultrafilter on $S$ which does not have a cluster point.

Then $\FF$ 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.