The meaning of "believe" in the Bible is pretty straightforward, but runs deeper than what appears on the surface. When in the Old Testament people were called on to believe in Jehovah, it meant a belief that Jehovah would lead them in spiritual things, not just natural ones, that following Him would lead to spiritual life. When people believed in Jesus in the New Testament, it meant believing in Him fully, as God Himself, all-powerful. Implied in this level of belief also was belief in what He taught and the importance to follow His precepts. That's why Peter sank while walking on the water; he began to see the human element of Jesus rather than God himself. It's also why Jesus could not do miracles in his home area; people there had known Him as a boy and young man, and could not believe in Him as God.

(Notae: Arcana Coelestia 6956, 6969, 6970, 7065, 8240, 8783, 10392; The Apocalypse Explained 815 [2-15])