An
epistemological theory of the way
beliefs are
justified. Under foundationalism a belief is justified if it is based on sound
basic beliefs. These are the 'foundations' of
knowledge from which
justification can flow.
When making a statement, one can always be questioned on 'why do you think that?'. And the reasons given can in turn be questioned. And so on. This could go on forever. It could also turn out that you really don't have any good reason to believe x after all. Or there could be something that you can be sure about. These 'certain things', if they can be found, would be your basic beliefs.
Strong Foundationalism says that you need to have basic beliefs that are absolutely infallible, incorrigible, or indubitable. But since it is very hard to find beliefs that are that strong, we also have Moderate Foundationalism, in which the basic beliefs only have to be 'most likely' true. Under this system you are justified in believing something only if it has a greater than 50% chance of being so. But even this can be hard to get, so we also have Weak Foundationalism--If you can get any justification at all, go ahead and use it.
Needless to say, people are having a hard time figuring out what we are justified in believing. Foundationalism is only one theory. You might also want to check out Coherentism, Reliabilism and Scepticism.
If you are looking for more information (and there is a lot more to foundationalism than I have here) Here is a good philosophy search engine. And any introduction to epistemology book should have a section on foundationalism.