Given an element x∈X, construct the set of all sets containing x:

E = Ex = {A⊆X | x∈A}

E is known as a principal ultrafilter.

Then it is easy to see that E is a filter (see cjeris' writeup!).

And if F!=E were a filter containing E, then F would contain some set B not containing x and also {x}, so it would contain their intersection -- the empty set. That's a contradiction, so E is maximal, hence an ultrafilter.

Principal ultrafilters are the only constructible ultrafilters.