What does Acts 11:13 mean?
The story of Cornelius and the angel is important. All Scripture is meaningful and preserved for a reason (2 Timothy 3:16–17). However, this particular story is recounted three times in the Word of God. Peter is in Jerusalem relating how a centurion named Cornelius had told him about his vision in Caesarea Maritima. Cornelius is Roman but worships the Jewish God and is well respected for his giving nature and his constant prayer. An angel came and told him to call for Peter, who was thirty-five miles south in Joppa, to hear what he had to say (Acts 10:3–6). He did so, Peter came, and Cornelius told him about the angel's message (Acts 10:30–32).Peter is telling Cornelius' story to the leaders of the church in Jerusalem. Some of the "circumcision party"—Jewish Jesus-followers who are still legalistic regarding the Mosaic law—have condemned Peter for eating with Gentiles. Peter is explaining why he did so: Jesus sent him a vision, the Holy Spirit told him to go, and an angel told Cornelius to send for him (Acts 11:1–12). That's a pretty strong defense.
Peter adds more of the angel's message than what is recorded in the original story in Acts 10. The angel tells Cornelius that Peter will explain how Cornelius and his household can be saved (Acts 11:14). It's very possible that at this time, Philip the evangelist is in Caesarea (Acts 8:40). But sharing Jesus' story "to the end of the earth" is the responsibility of the twelve apostles (Acts 1:8). It was Peter who preached to the Greek-cultured Jews on Pentecost (Acts 2) and who, along with John, validated the faith of the Samaritans with whom Philip shared the gospel (Acts 8:14–17). So, it is Peter who brought the salvation of Jesus to a houseful of Gentiles.