"'We are witnesses of everything he did in the country of the Jews and in Jerusalem. They killed him by hanging him on a tree, but God raised him from the dead on the third day and caused him to be seen. He was not seen by all the people, but by witnesses whom God had already chosen- by us who ate and drank with him after he rose from the dead.'" NIV translation
In this passage, Peter is giving his credentials to the people at Cornelius' house. He is reminding them that God had a plan and that, even after the death of Jesus, God was still in control. He reminds them that God raised Jesus from the dead and that he appeared to a select group of people of whom Peter is one.
"'He commanded us to preach to the people and to testify that he is the one whom God appointed as judge of the living and the dead. All the prophets testify about him that everyone who believes in him receives forgiveness of sins through his name.'" NIV translation
Now, Peter gets down to it. In the previous passage, he gives his qualifications as an apostle, now, he explains his purpose in coming to Cornelius' house. That purpose (which is the same as our purpose as Christians) is to testify about Jesus Christ. He explains that Jesus will be the judge of all mankind and that forgiveness comes through him.
"While Peter was still speaking these words, the Holy Spirit came on all who heard the message. The circumcised believers who had come with Peter were astonished that the gift of the Holy Spirit had been poured out even on the Gentiles. For they heard them speaking in tongues and praising God." NIV translation
What a powerful example of why we should share the Word of God. Here we see how God can immediately take those words and change people's lives. The believers that were with Peter also got to see that God loves all men. We probably all at one time have seen someone so far from God that we did not see how they could possibly be reached. This same feeling is what the Jewish believers felt but they got to see God in action. Imagine the shock when they saw God touch the lives of these untouchable people. The same thing can and will happen to the "untouchables" in our paths.
"Then Peter said, 'Can anyone keep these people from being baptized with water? They have received the Holy Spirit just as we have.' So he ordered that they be baptized in the name of Jesus Christ. Then they asked Peter to stay with them for a few days." NIV translation
Here, Peter is asking a silly question and answering it at the same time. In the previous passage, the people had received the gift of the Holy Spirit. You see, the Holy Spirit is our counselor that constantly guides us to please God and it pleases God when we obey. Part of that obedience is being baptized (just like Jesus did). Peter knew that now they were going to want to please God and so they were baptized in the name of Jesus Christ.