1....This Psalm is one of those promises, surely and openly prophesying our Lord and Saviour Jesus Christ; so that we are utterly unable to doubt that Christ is announced in this Psalm, since we are now Christians, and believe the Gospel. For when our Lord and Saviour Jesus Christ asked of the Jews, whose Son they alleged Christ to be, and they had replied, “the Son of David;” He at once replied to their answer, “How then does David in spirit call Him Lord, saying, The Lord said unto My Lord?” etc. “If then,” He asked, “David in the spirit call Him Lord, how is He his son?” With this verse this Psalm begins.
2. “The Lord said unto my Lord, Sit on My right hand, until I make Your enemies Your footstool”. We ought, therefore, thoroughly to consider this question proposed to the Jews by the Lord, in the very commencement of the Psalm. For if what the.Jews answered be asked of us, whether we confess or deny it; God forbid that we should deny it. If it be said to us, Is Christ the Son of David, or not? If we reply, No, we contradict the Gospel for the Gospel of St. Matthew thus begins, “The book of the generation of Jesus Christ, the Son of David.” The Evangelist declares, that he is writing the book of the generation of Jesus Christ, the Son of David. The Jews, then, when questioned by Christ, whose Son they believed Christ to be, rightly answered, the Son of David. The Gospel agrees with their answer. Not only the suspicion of the Jews, but the faith of Christians, does declare this....“If then David in the spirit called Him Lord, how is He his son?” The Jews were silent at this question: they found no further reply: yet they did not seek Him as the Lord, for they did not acknowledge Him to be Himself that Son of David. But let us, brethren, both believe and declare: for, “with the heart we believe unto righteousness: but with the mouth confession is made unto salvation;” let us believe, I say, and let us declare both the Son of David, and the Lord of David. Let us not be ashamed of the Son of David, lest we find the Lord of David angry with us.
3....We know that Christ sits at the right hand of the Father, since His resurrection from the dead, and ascent into heaven. It is already done: we saw not it, but we have believed it: we have read it in the Scripture, have heard it preached, and hold it by faith. So that by the very circumstance that Christ was David's Son, He became His Lord also. For That which was born of the seed of David was so honoured, that It was also the Lord of David. You wonder at this, as if the same did not happen in human affairs. For if it should happen, that the son of any private person be made a king, will he not be his father's lord? What is yet more wonderful may happen, not only that the son of a private person, by being made a king, may become his father's lord; but that the son of a layman, by being made a Bishop, may become his father's father. So that in this very circumstance, that Christ took upon Him the flesh, that He died in the flesh, that He rose again in the same flesh, that in the same He ascended into Heaven, and sits on the right hand of His Father, in this same flesh so honoured, so brightened, so changed into a heavenly garb, He is both David's Son, and David's Lord....
Source: The Enarrations, or Expositions, on the Psalms (New Advent)