The ultrafilter theorem is the statement that any filter on a set can be extended to an ultrafilter. It is perhaps more common to see it sated as the (Boolean) Prime ideal theorem: Every Boolean algebra admits a prime ideal.

The Hahn-Banach theorem is actually equivalent to the statement that every Boolean algebra admits a real-valued measure, but this is not entirely straightforward (see Luxemburg, "Reduced powers of the real number system and equivalents of the Hahn-Banach extension theorem", Intern. Symp. on the applications of model theory, (1969) 123-127).

For a discussion of Hahn-Banach vs. Choice and some additional remarks and references, see Jech "The axiom of choice", North-Holland, 1973.

You may also be interested in the references I include in this answer.

The ultrafilter theorem is the statement that any filter on a set can be extended to an ultrafilter. It is perhaps more common to see it sated as the (Boolean) Prime ideal theorem: Every Boolean algebra admits a prime ideal.

The Hahn-Banach theorem is actually equivalent to the statement that every Boolean algebra admits a real-valued measure, but this is not entirely straightforward (see Luxemburg, "Reduced powers of the real number system and equivalents of the Hahn-Banach extension theorem", Intern. Symp. on the applications of model theory, (1969) 123-127).

For a discussion of Hahn-Banach vs. Choice and some additional remarks and references, see Jech "The axiom of choice", North-Holland, 1973.

You may also be interested in the references I include in this answer.