Everyone has things that he or she thinks are true. For example, you may think that there is a computer in front of you right now. Maybe you are not on a computer, but access the Internet through a mobile device. You may even be reading this after having printed it. None of that matters. The point is that you believe you are reading this right now. If I were to ask you about why you believe this, you would appeal to something else to explain your belief. Perhaps you would mention that you see the text in front of you. If you were pressed further, you might mention that you believe you see the text because you trust your senses to be reliable. You may be pushed for a further response and say that you think your mind is able to correctly interpret what your senses are telling it.
So, if I were to ask you about C, you would appeal to B. If I wanted to know why you believed B, then you would appeal to A. This could go on for a long time, but it cannot go on forever. Everyone has to stop somewhere. Eventually you will reach your highest authority and be unable to go any further back, just like the situation with the Supreme Court. There is always a highest court of appeals.
Some people take their highest court to be their own reason; their thought processes are said to be the final authority. Why do these people take reason to be their highest authority though? Is it because their reason tells them to? This, of course, is circular reasoning. This final authority, or highest court of appeals, is accepted on faith.
Some people take their highest authority to be the senses. If the highest court of appeals is the senses, then there is nothing else to appeal to higher than the senses. The reliability of the senses will have to be accepted on faith.
The same holds true for every final authority. There are some things we have to believe before we can make sense of anything else. Some people accept their reasoning on faith. Some people accept their senses on faith. There are other positions as well. Some people say that they reject all of these final authorities, which dumps them into the endless chasm of skepticism or subjectivism. These people claim to know nothing at all, or claim that they know whatever they feel like knowing. Both of these positions obviously have some very serious problems. The most significant problem with them is that people are able to function in the world and to make sense of things. This means that ultimately, people have a highest court of appeals, a final authority, a presupposition.
As Christians we take as our presupposition the truth that God has revealed Himself to us through the Christ of Scripture. From this faith commitment we are able to derive an entire system of knowing other things. We are not just limited to what can be known in Scripture, but rather are able to take the principles taught in Scripture as a basis for knowing everything else we are able to know as well. Rejecting the Christian presupposition is placing oneself above God and results in foolishness. (Psalm 14.1; Proverbs 1.7) There are no successful objections to the Christian faith when it is properly understood and taken as a whole. Meanwhile, the rejection of Christian presuppositions results in self-defeating skepticism or subjectivism as the history of philosophy has adequately shown.
How can people who claim that God does not exist still know things then? How can non-Christians make sense of things? What about very intelligent non-Christians? What about all the good things non-Christians have been able to bring about for society? What sort of answers could we possibly conjour up in response to these probing questions?
This, I think, is the presuppositional understanding of things in a nutshell.