Perhaps the best way of understanding what Jesus meant by the "kingdom of heaven" is to look at what he taught about it. And, to find out what Jesus taught, we must turn to his parables.
MT 13:34 "Jesus used parables to tell all these things to the crowds; he would not say a thing to them without using a parable"
Comment: That Jesus would use parables to teach the masses of humans, but explain these to the apostles has led some to claim that Jesus had "two" messages: one for the common people, and another "special" message for his chosen few. Most Christians in the past have rejected this, but one can understand how the following passage in Matthew could lead to this misunderstanding: "Then the disciples came to Jesus and asked him, `Why do you use parable when you talk to the people?' Jesus answered`The knowledge about the secrets of the Kingdom of heaven has been given to you, but not to them.' (MT 13:10-11)" In addition, Mark's gospel states "He [= Jesus] would not speak to them [= the people] without using parables, but when he was alone with his disciples, he would explain everything to them. (MK 4:34)"