Edge (the magazine, not the guitarist) likes to postulate a question each year to a group of prominent thinkers (scientists, philosophers, etc.). One year, the question was, "What do you believe is true, even though you cannot prove it?"
This is a great question because it allows space for something that we intuitively "know" even without any evidence one way or the other. Perhaps the most obvious answers were those that involved God. Although many great minds have attempted to argue for or against the existence of God, none have yet been able to actually prove their case.
Such a result does not necessarily mean that the existence of God is only in the realm of intuition. If that were true, neurobiologists would still be looking for the so-called "God spot" in the brain. Such a search was abandoned because the spot doesn't exist. The question of the existence of God should be considered as a matter of objective fact about the universe. Either God exists, or God does not exist. The manner and mode of God's existence is something that can be argued about by those philosophers and theologians who are willing to believe without proof that God exists. Those who believe without proof that God does not exist are quite unlikely to get involved in the debate.
So, to believe without proof is something that we all do, perhaps without realising it. Some call this activity "faith" but faith is such an emotionally and socially charged word that I will avoid using it where possible. The breadth of meaning for the word is too wide for it to be useful. Instead, to believe is merely shorthand for "believe without proof for or against."
Belief without proof is also different to belief without reason. Most people who believe something is true will have a good reason for it. A person who has dreamt about their child saying a specific thing prior to the child even being conceived, and then hearing the child say that very thing when old enough to talk... such a person will have reason to believe that there is something more to the universe than merely the 4 dimensions of space-time. This is belief with reason.
In a way, most reasons are experiences. Someone with a better grasp of poetic English than me once said that the person with an experience is never at the mercy of the person with an argument. A rational argument can batter against an experience until the end of time and have no effect on the belief that comes from that experience. Ultimately, experience - and the conclusions that we make from those experiences - will shape our beliefs more than any logical or consistent argument.
The only hope for someone who is trapped in a false belief is to investigate how they went from the experience to the belief. This is the niche of the structuralists and it is at this point that I tip my hat to Žižek and his crowd for their relentless project to find holes in the ordinary experiences of humanity in order to force those who are trapped in them to question the world in which they exist. In other words, question everything around you, especially the things you think don't need to be questioned.
Belief is not the enemy. Without belief, there would be no organised enquiry to find the truth. At present there are a number of people who believe that the Higgs boson will be observed. There is no proof that it exists, but there is a lot of belief and a great deal of reason. Belief is quite useful. It will direct you towards the truth.
So what do I believe that I cannot prove? I believe that there is a God and that this God probably did not create the universe. The rest of that argument will have to wait for another time.