Eight days later was the first day of the Jewish week (Sunday). This was one week after the resurrection of Jesus. At that time, people included the current day when counting forward. The appearance of Jesus on this Sunday helps explain why the disciples met for worship on "the Lord's Day" (see Revelation 1:10).
Peace be with you: This scene was just like the first appearance of Jesus (John 20:19–20). Jesus had already heard the complaint of Thomas, and now Jesus answered directly.
believe: Jesus challenged Thomas to believe in the resurrection like the others did.