Peter again quotes one of David's Psalms. David is not speaking of himself in this Psalm. "The Lord" and "my" refer to God; "my Lord" and "your" refer to Jesus the Messiah.
To sit at the "right hand of God" is a symbolic action of receiving great honor and authority from God. Alternate translation: "Sit in the place of honor beside me" (See: [[rc://en/ta/man/translate/translate-symaction]])