In Acts 2, Peter has been making an argument for establishing the identity of Jesus as both Lord and Messiah. His argument continues in Acts 2:33–35 where he quotes Psalm 110:1:
Therefore having been exalted to the right hand of God, and having received from the Father the promise of the Holy Spirit, He has poured forth this which you both see and hear. For it was not David who ascended into heaven, but he himself says: “THE LORD SAID TO MY LORD, ‘SIT AT MY RIGHT HAND, UNTIL I MAKE YOUR ENEMIES A FOOTSTOOL FOR YOUR FEET.’”
Then immediately in 2:36 Peter makes his closing argument: “Therefore let all the house of Israel know for certain that God has made Him both Lord and Christ—this Jesus whom you crucified.”
Peter had already declared that David was a “prophet” (2:30) and that David “looked ahead” to Christ (2:31). So when Peter quoted Psalm 110:1 in reference to Jesus, he clearly viewed Jesus as the literal fulfillment of what Psalm 110:1 says.
Peter’s use of Psalm 110:1 in Acts 2:34–25, therefore, is a case of direct literal fulfillment of an OT prophetic/messianic text. While it appears fashionable in more recent times to view none of the psalms as explicitly messianic, Psalm 110 is a clear instance where David himself had the coming Messiah explicitly in mind when he wrote the psalm. The reasons for this are several.
First, Jesus viewed Psalm 110 as a case where David was referring to Jesus. In Matt 22:41–46, Jesus stumped the Pharisees by showing that the Messiah was both the son of David and David’s Lord. Thus, this figure that David spoke of was both human and divine. Jesus viewed Psalm 110 as messianic and referring to himself.
Second, the figure in Psalm 110 is an eternal priest-king (Psalm 110:1, 4) like Melchizedek, the priest-king. This "forever" aspect fits the picture of the coming Messiah, Jesus, better than David.
Third, Peter, in Acts 2:33–35, explicitly states that David viewed Jesus’ exaltation to the right hand of God as being the fulfillment of what David meant in Psalm 110:1. This is the case because the second “Lord” of Psalm 110:1 is at the right hand of God sharing Yahweh’s authority. But David, unlike Jesus, never ascended to the right hand of God. So like Acts 2:25–28 and its use of Psalm 16:8–11, we have an explicit case of inspired commentary from an apostle (Peter) on what an OT author (David) meant.
In sum, Acts 2:33–35 is a case of direct literal fulfillment of an OT prophetic text. Again, we see another contextual use of the OT by the NT persons.