I have seen the use of ultrafilters in topology to prove Tychonoff's theorem. In fact, this is the only application of ultrafilters in topology I have been able to find so far.
Are there any other applications of ultrafilters to topology?
http://ncatlab.org/nlab/show/ultrafilter will give you a lot of uses of ultrafilters, including in topology.
Two things that certainly must be noted are the classical results that the category of compact Hausdorff spaces and that of all topological spaces arise as categories of algebras for suitable ultrafilter monads.