"Repent, then, and turn to God, so that your sins may be wiped out, that times of refreshing may come from the Lord, and that he may send the Christ, who has been appointed for you- even Jesus." NIV translation
After spending quite a bit of time reasoning with the crowd and letting the Holy Spirit convict them, Peter now turns to the good news. He tells them that, if they repent, God has appointed the same Jesus that they turned over for crucifixion as their messiah. You see, the Jews believed in a messiah they just did not believe that it was Jesus and they thought that the messiah would immediately usher in a new Jewish kingdom. Peter tells them that this messiah would wipe away their sins.
"He must remain in heaven until the time comes for God to restore everything, as he promised long ago through his holy prophets." NIV translation
Now that Peter has explained that Jesus is the Savior, he starts to explain why the Messiah is not what they expected. You see, the Jews knew the Old Testament and that the prophets had foretold the coming of the Messiah (Savior) but it was not what they had expected. The Jewish people were expecting the Messiah to come and immediately restore the earthly kingdom of Israel.
This verse also contains a promise for us today. That promise is that God is going to restore everything someday. The earth will be made new and we will get to be with God just as Adam and Eve were in Eden before their sin.
"For Moses said, 'The Lord your God will raise up for you a prophet like me from among your own people; you must listen to everything he tells you. Anyone who does not listen to him will be completely cut off from among his people.'" NIV translation
Peter now goes back to using the Old Testament to get the crowd to understand who Jesus is and what it means. He quotes Moses which the Jewish people respect as the one who brought them the law. Peter points out that this same Moses predicted that there would be a prophet after him that they must listen to. He goes on to say that this prophet will come from the Israelite people. If you look at the beginning of the Gospel account of Matthew, you notice that the first part is tracing the lineage of Jesus back to the Israelite kings. After pointing out that, according to Moses, the prophet would come, Peter then tells them that if they do not listen to him, they will be cut off from among his people. That is also important to us in the world today. It is popular today to say that there are many ways to have a relationship with God. This passage of scripture goes back to the beginning and makes it known that Jesus is the only way.