In Acts 3, what kingdom was Peter talking about?
Peter was speaking of the kingdom promised to Abraham, an earthly messianic kingdom. It would be established when Israel repented and believed Jesus was Messiah, and God sent Christ back to earth. This kingdom would first last a thousand years, though the thousand-year timeframe was not revealed until Revelation.
Les explains that in Acts 3 Peter's thought continues from verse 19 into verse 20—a semicolon, not a period—so that if Israel repented, God would send refreshing in the person of the Lord Jesus Himself. He would send Jesus Christ, who was before preached. Les notes that if Christ is to be sent back to earth, the time element was kept secret until revealed to John in Revelation. When Christ returned, this kingdom would become viable and visible, not immediately eternal, but first lasting a thousand years. Les emphasizes that nothing in Scripture indicates that thousand-year time frame until Revelation. Peter was speaking of the kingdom promised all the way back to Abraham: when their Messiah would return based on Israel's repentance and believing who Jesus is, then God would have sent Christ in verse 20. But verse 21 required a seven-year interlude Daniel spoke of in chapter 9—the seven years of tribulation—which is the times of restitution spoken of here.