It is not hard for us to appreciate that Jesus was not primarily interested in what other people though about him. What was of interest to him was a public confession of him by his disciples. And one was given by Peter when he said who he thought Jesus was. His answer reveals that he had grasped two important details about Jesus.
The first detail that Peter confesses is that he believes that Jesus is the Christ. In using this name, Peter was stating that he believed that Jesus was the promised Messiah, which causes us to think back to how Andrew described Jesus when he witnessed to his brother for the first time about Jesus. This confession does not mean that Peter understood everything about Jesus because shortly after he would object to the statement that Jesus would be arrested and put to death (Matt. 16:21-23). Yet he had grown in his learning about Jesus.
We can see more of that growth in the second title that Peter uses when he confesses that Jesus is the Son of the living God. Other passages indicate that for someone to claim to be the Son of God was a claim to deity, of being equal with God. So we can see that Peter has made real progress in his understanding of who Jesus is. As a Jew, Peter would have held tenaciously to the doctrine that God is one, but while still holding to that belief here he is stating that he now believes that there is more than one person in the Godhead.
Some may wonder why Peter wanted to describe God as the living God. One reason might be their location. Caesarea Philippi was near to pagan temples and their existence could have caused Peter to be thankful that he knew that the true God was not like the idols connected to those temples.
The titles that Peter uses remind us of the greatness of Jesus. Peter was discovering more and more about how great Jesus is, even although he had a lot more to discover. And he is still discovering how amazing Jesus is.