Applications
Intuitively, the Boolean prime ideal theorem states that there are "enough" prime ideals in a Boolean algebra in the sense that we can extend every ideal to a maximal one. This is of practical importance for proving Stone's representation theorem for Boolean algebras, a special case of Stone duality, in which one equips the set of all prime ideals with a certain topology and can indeed regain the original Boolean algebra (up to isomorphism) from this data. Furthermore, it turns out that in applications one can freely choose either to work with prime ideals or with prime filters, because every ideal uniquely determines a filter: the set of all Boolean complements of its elements. Both approaches are found in the literature.
Many other theorems of general topology that are often said to rely on the axiom of choice are in fact equivalent to BPI. For example, the theorem that a product of compact Hausdorff spaces is compact is equivalent to it. If we leave out "Hausdorff" we get a theorem equivalent to the full axiom of choice.
A not too well known application of the Boolean prime ideal theorem is the existence of a non-measurable set (the example usually given is the Vitali set, which requires the Axiom of Choice). From this and the fact that the BPI is strictly weaker than the Axiom of Choice, it follows that the existence of non-measurable sets is strictly weaker than the axiom of choice.
Read more about this topic: Boolean Prime Ideal Theorem