Foundationism is the position that a belief is justified if it can be inferred or derived from some set of ‘foundational’ or ‘basic’ beliefs, which themselves do not require any further justification. Such inferences from the foundational believed need not be direct: belief A could be justified by belief B, which in turn could be justified by belief C, and so on, as long as eventually the chain of justification ends with some belief which is one of the special foundational beliefs requiring no further justification. By proposing a termination to the chain of justifications, foundationalism is one possible response to the regress argument.
There are a number of explanations foundationalists typically give as to why the foundational beliefs, unlike all others, do not require justification. One approach is to say that such beliefs are self-evident, meaning that anyone who truly understands them will necessarily believe them – for example, a statement like “two plus two equals four” is sometimes said to be self-evidently true to anyone who understands its meaning. A slightly different approach is to say that basic beliefs are incorrigible, meaning that they cannot be doubted. These and other approaches have all been subject to considerable critique, and explaining how basic beliefs can be justified while defending against the charge that basic beliefs are merely arbitrary, is among the foremost ongoing challenges for foundationalists.
Foundationalism: introductory piece from TheoryOfKnowledge.info
Foundationalism: more detailed analysis from the Internet Encyclopedia of Philosophy
Foundationalist theories of epistemic justification: a detailed overview of classical and modern approaches to foundationalism from the Stanford Encyclopedia of Philosophy