What does Acts 10:40 mean?

Peter is speaking to a houseful of Gentiles in Caesarea Maritima about Jesus. They know vaguely who He is, and they follow the Jewish God, but they don't know the specifics. Because they are not Jews, like Stephen's audience (Acts 7), Peter does not go into detail about how Jesus fulfills the Old Testament prophecies, although he does mention it (Acts 10:43). Instead, he starts with John's baptism—the public admission of sin and choice to repent and follow God more faithfully. It's not stated if Peter's host, a Roman centurion named Cornelius, knew John, but he does know of him and seems to follow his teaching (Acts 10:1–2, 37).

When Pilate was governor of Judea, he seemed to spend much of his time in Jerusalem. He was removed from his position and Herod Agrippa I was given the title of king of Samaria and Judea. He kept his capital in Caesarea, on the coast of the Mediterranean. As a soldier, Cornelius would have heard the rumors surrounding Jesus' death. The story of what happened after is more nebulous.

In Matthew 27:62–66, the chief priests and Pharisees approach Pilate, requesting security measures for Jesus' tomb so the disciples can't steal His body and claim He rose from the dead. When the angels arrive, the guards prove ineffective (Matthew 28:4). The guards report back to the priests that Jesus is gone; the priests pay them off, telling them to say they fell asleep and the disciples stole the body (Matthew 28:11–15).

Tradition says these guards were Roman soldiers. Cornelius certainly would have heard something about the account. For one, news spreads quickly and widely in the military. For another, if a Roman soldier fell asleep during guard duty, he was vulnerable to execution. But it's not clear what Cornelius heard about the events after Jesus' crucifixion—probably rumors and half-truths that Peter is now clearing up.