The Wikipedia explanation:
Foundationalism is about knowledge that ultimately rests on axioms, i.e. basic beliefs or first principles. The admiral ship of foundationalism is, beyond any doubt, mathematics.Foundationalism concerns philosophical theories of knowledge resting upon justified belief, or some secure foundation of certainty such as a conclusion inferred from a basis of sound premises. The main rival of the foundationalist theory of justification is the coherence theory of justification, whereby a body of knowledge, not requiring a secure foundation, can be established by the interlocking strength of its components, like a puzzle solved without prior certainty that each small region was solved correctly.
Identifying the alternatives as either circular reasoning or infinite regress, and thus exhibiting the regress problem, Aristotle made foundationalism his own clear choice, positing basic beliefs underpinning others.
In Aristotle's time, the flagship of foundationalism was (classical) Euclidean geometry.
In modern times, the dominant axiomatization is either Arithmetic Theory (PA) or Set Theory (ZFC).
Classical geometry is still a valid theory but considered to be quite weak.
The noisy cabal of set theorists will say that the most dominant axiomatic theory is Set Theory (ZFC), but everybody else will say that it is Arithmetic Theory (PA).
Aristotle believed that he would be able to reorganize all knowledge with foundationalist foundations. Nowadays, we know that this is not possible, and probably, not even desirable.