What does Jesus mean when He says that we should be like children? Matthew 18:1-4:

At that time the disciples came to Jesus and asked, “Who is the greatest in the kingdom of heaven?”
He called a little child and had him stand among them. And he said: “I tell you the truth, unless you change and become like little children, you will never enter the kingdom of heaven. Therefore, whoever humbles himself like this child is the greatest in the kingdom of heaven.

What does this mean? How can we apply it?
Great question!
There are so many things about this verse that are so awesome. I would propose Jesus' answer has a two-fold answer. 

Part 1
First, Jesus says that we will not enter the kingdom of heaven unless we become like a child. We need to change our mindset and think like children think. Not that we become immature and act childish in the traditional sense, but that we accept things in simplicity. Children have a way of believing everything you say and believing it as fact. It's only when we grow up that we tend to question everything. The currency of the kingdom of heaven is faith. It's what moves God's heart and what the great people of God are known for. So for the first part Jesus is telling us, "have faith like a child".

Part 2
The humility of children is very evident. It's a beautiful thing to watch. Although they may compete with each other in games and such, the humiliity they have without even knowing it is astouding! They were also thought of in Jewish culture as insignificant until they were 12. Essentially, Jesus is saying act as if you were the lowest in society. Jesus often tell us to go lower if we want to be higher. Our humility in His eyes equals in direct proportion to how He exalts us. The word says over and over that our job is to humble ourselves in His sight and He exalts us.

I hope this makes sense and helps :)