When we say we 'believe in' an idea, we usually mean that the idea is true or that it is a good idea. For example, 'I believe that life is sacred,' or 'I believe in the legal protection of life from conception to natural death.'
On the other hand, when we say we 'believe in' a person, we express a certain faith in him or her. It is a sort of confession of our belief that someone has the resources to succeed in some task or to be the person he or she is called to be at a certain moment. It often happens that someone struggling with self-doubt is encouraged by those who say, 'I believe in you.'
Belief 'in God' belongs in the second category, but is far too often expressed in the first. Even the question, 'Do you believe in God?' is usually to ask whether or not we believe in the 'existence of God.' But to actually believe in the existence of God is to confess the surpassing import of God; otherwise we are not believing in a god that it is really God. As Abraham Heschel famously put it, "God is no importance unless He is of supreme importance."
To believe in God is to believe in him in the same way we say we believe in another person. It is the confession that God has the power and will to carry out the promises he has revealed to us. As Isaiah the prophet puts it to us during Lent:
For just as from the heavens the rain and snow come down And do not return there till they have watered the earth, making it fertile and fruitful, Giving seed to him who sows and bread to him who eats, so shall my word be that goes forth from my mouth; It shall not return to me void, but shall do my will, achieving the end for which I sent it. (Isaiah 55:10-11)