What does Acts 10:43 mean?

Peter is in Caesarea Maritima, the Roman capital of Samaria and Judea, at the home of Cornelius, a Roman centurion who devoutly follows the Jewish God. An angel appeared to Cornelius and told him to send for Peter. As his messengers approached the house where Peter was staying, Jesus sent Peter a vision explaining that Gentiles were welcome to follow Him, as well. Peter went with the messengers and found a household of people ready to hear his testimony of Jesus (Acts 10:1–33).

The "prophets" are those mentioned in the Old Testament, including Moses (Deuteronomy 18:15), David (Psalm 22), Isaiah (Isaiah 53; 61:1–3), Hosea (Hosea 6:1–2), and Jonah. They, like the apostles (Acts 1:8), bear witness to Jesus' identity as the Son of God and the Jewish Messiah. Jesus "name" doesn't mean the syllables Peter uses to identify Him. It means His power, authority, sovereignty, and deity: His identity. Jesus saves because of who He is.

"Forgiveness" is from the Greek root word aphesis. This refers to when a victim voluntarily releases their right to personally punish their aggressor. The governing authorities will still hold the perpetrator legally and financially responsible for their crime, but the impetus comes from the law, not the victim. Jesus covered the legal ramifications of sin at the cross, and offers both personal and legal forgiveness for any who trust in His name.