The text from Acts 2:34 is a quotation from Psalm 110:1.
In Hebrew:
מזמור לדוד (A Psalm of David) נאם (The declaration of) יהוה (the LORD) לאדני (to my Lord), שׁב (Sit) לימיני (at my right hand), עד (until) אשׁית (I make) איביך (your enemies) הדם (your footstool). לרגליך (under your feet).
In Greek:
εἶπεν (said) ὁ (the) κύριος (Lord) τῷ (to the) κυρίῳ (Lord) μου (of me), κάθου (Sit) ἐκ (at) δεξιῶν (right hand) μου (of me), ἕως (until) ἂν (I make) θῶ (your) τοὺς (the) ἐχθρούς (enemies) σου (of you) ὑποπόδιον (a footstool) τῶν (of the) ποδῶν (feet) σου (of you).
Jesus provided a key insight into interpreting these verses when he mentioned that David spoke them "in spirit," signifying a trance-like state (Matthew 22:43).
Matthew 22:43
λέγει αὐτοῖς• πῶς οὖν Δαυὶδ ἐν πνεύματι καλεῖ αὐτὸν κύριον λέγων•
He saith unto them, How then doth David in spirit call him Lord, saying
In Psalm 110:5, David is no longer in a "spirit" state, and thus, he sees the Messiah merely as his carnal descendant, one who would come from his loins. This is why the Messiah, in his carnal form, is "at his right hand."
Psalm 110:5 (KJV):
"The Lord at thy right hand shall strike through kings in the day of his wrath."
Thus, the entire Psalm is in the third person and sung by the Levites, indicating that the Messiah will not be merely a carnal man but an eternal Elohim King and Priest.