"Therefore my heart is glad and my tongue rejoices; my body also will live in hope, because you will not abandon me to the grave, nor will you let your Holy One see decay." NIV translation
Peter is still quoting King David here and it is a very good model for us. David says, in the first part of the quote that his "heart is glad and my tongue rejoices". He says this because of the verse before and the fact that the Lord is always with him. He goes on to say that he will live in hope because of the promise of not being abandoned to the grave. We need to ask ourselves every day if we are living in hope and rejoicing. If we are doing that, then those around us will be able to see Jesus through us and that is what it is all about. David finishes off this verse by saying that God would not let his Holy One see decay. That is what has happened not long before Peter addressed this crowd of Jews. Jesus died on the cross for our sins but did not remain in the tomb to see decay. He was resurrected and Peter is using King David to make them see that, through Jesus, these words have been fulfilled.
"You have made known to me the paths of life; you will fill me with joy in your presence." NIV translation
This is the final part of Psalm 16 that Peter quotes to the crowd. In this verse, King David is rejoicing that God has been with him and guided him. Then, David looks forward to the time when he will be filled with joy in the presence of the Lord. That brings up a good question for us, as Christians. Are you living with joy?
You see David had to look forward to the time when he would be with God but we have God with us every day. The Holy Spirit is God and he is inside of each one of us if we are Christians. That is what sets us apart and should make us have joy. No matter what happens, we still have God in us and with us to get us through. That, my friends, should bring us joy.
"Brothers, I can tell you confidently that the patriarch David died and was buried, and his tomb is here to this day. But he was a prophet and knew that God had promised him on oath that he would place one of his descendents on his throne. Seeing what was ahead, he spoke of the resurrection of the Christ, that he was not abandoned to the grave, nor did his body see decay." NIV translation
Peter has finished his quote of David in the Psalms and now starts to reason with this crowd of mostly Jews. Peter starts to contrast King David with Jesus by referring that David had died and that you could go and find his body in the grave. Then he contrasts that with Jesus and the fact that the tomb they layed him in was empty (Jesus was resurrected) and the fact that, unlike David, you could not go see Jesus' body in a tomb where it was decaying. What's more, Peter points out that King David had foreseen this and wrote about it in the Psalm that was quoted.