Excerpt
Filters (and ultrafilters) are used in various areas of mathematics, e.g. general topology, set theory, boolean algebras, model theory.
In topology they can be used to study convergence in a more general way than just convergence of sequences.

Wiki
[Filters](http://en.wikipedia.org/wiki/Filter_%28mathematics%29) (and [ultrafilters](http://en.wikipedia.org/wiki/Ultrafilter)) are used in various areas of mathematics, e.g. general topology, set theory, boolean algebras, model theory.
In topology they can be used to study convergence in a more general way than just convergence of sequences.
Various types of ultrafilters (such as good ultrafilters, Ramsey ultrafilters, P-points) are useful in set theory and model theory and their applications.