That oversized mess of a word means just what it sounds like: the believer argues that one must presuppose the existence of God in order to have any kind of valid epistemology at all. This isn't some fringe idea at all – Frank Turek argued it recently in his debate with David Silverman (I suffered through some of it); it's the root of Alvin Plantinga's "evolutionary argument against naturalism"; and David Marshall, Tom Gilson and Peter Grice (among others) argued it in the apologetics essay collection "True Reason".
It seems like it should go without saying that these two routes are incompatible, though apologists rarely if ever get taken to task for the error. Think about it: you presuppose that God exists, and then the evidence you examine leads you to affirm God's existence. If that ain't circular reasoning, nothing is. If presuppositionalism is valid, then God's existence is self-evident. Apologists rarely if ever use this exact language (probably because it betrays how stupid presuppositionalism actually is), but it's the inevitable logical conclusion – God's existence can't be supported by evidence even in principle, because without God you couldn't have evidence in the first place.
A disturbing lack of alternatives
One of my all-time favorite posts from the mighty Sean Carroll is one called "Faith and Epistemological Quicksand". He didn't use the word, but it was actually an incisive rebuttal of presuppositionalism. He said (emphasis mine),
Any time we have beliefs of any sort, we need to admit the possibility that they are incorrect. Even if we have think that some result has been reached by nothing but the application of pristine mathematical logic (e.g. the ABC conjecture), it’s always possible that we simply made a mistake — have you ever multiplied two numbers together and gotten the wrong answer? Certainly in an empirical endeavor like science, we recognize that our theoretical understanding is necessarily contingent, and are constantly trying to do better, via more precise and far-reaching experimental tests. These are methods of reaching knowledge that have built-in methods of self-correction.
So what about faith? Even if your faith is extremely strong in some particular proposition, e.g. that God loves you, it’s important to recognize that there’s a chance you are mistaken. That should be an important part of any respectable road to knowledge. So you are faced with (at least) two alternative ideas: first, that God exists and really does love you and has put that belief into your mind via the road of faith, and second, that God doesn’t exist and that you have just made a mistake.
The problem is that you haven’t given yourself any way to legitimately decide between these two alternatives. Once you say that you have faith, and that it comes directly from God, there is no self-correction mechanism. You can justify essentially any belief at all by claiming that God gave it to you directly, despite any logical or evidence-based arguments to the contrary.In a nutshell, that's the problem with presuppositionalism. If you presuppose that the existence of God is foundational to any valid epistemology, then you have no way of knowing whether you're right or wrong about your presupposition. All evidence doesn't merely lead to God's existence, but the mere conceptualization of evidence itself necessitates God's existence. That means there can be no self-correction mechanism – no methodology by which the assumption of God's existence could be falsified, since any rational examination of the evidence would include the a priori assumption of God's supposedly self-evident existence. Apologists deserve a gold medal for such sophistic gymnastics.
And when you think about it, isn't it pretty hilarious that one of the great triumphs of 'sophisticated' apologetics is that theologians have come up with an elaborate rationalization for assuming, a priori, precisely what they are trying to prove?