'Yahweh says to my lord'
A royal psalm's court setting, Jesus' riddle about David's son, the Melchizedek argument of Hebrews 7, and the rabbinic readings.
Psalm 110:1-4
1Yahweh says to my Lord, “Sit at my right hand, until I make your enemies your footstool for your feet.”
2Yahweh will send out the rod of your strength out of Zion. Rule among your enemies.
3Your people offer themselves willingly in the day of your power, in holy array. Out of the womb of the morning, you have the dew of your youth.
4Yahweh has sworn, and will not change his mind: “You are a priest forever in the order of Melchizedek.”
Matthew 22:41-46
41Now while the Pharisees were gathered together, Jesus asked them a question,
42saying, “What do you think of the Christ? Whose son is he?” They said to him, “Of David.”
43He said to them, “How then does David in the Spirit call him Lord, saying,
44‘The Lord said to my Lord, sit on my right hand, until I make your enemies a footstool for your feet?’
45“If then David calls him Lord, how is he his son?”
46No one was able to answer him a word, neither did any man dare ask him any more questions from that day forward.
What each tradition says
Rabbinic / Midrashic
Rashi explains Psalm 110:1 according to the Rabbis as referring to Abraham our father. “The word of the Lord to my master” is addressed to Abraham, whom the world called “my master,” as Genesis 23:6 says, “Hearken to us, my master.”
Rashi (Rabbi Solomon ben Isaac, 1040-1105), Rashi on Psalms 110:1, in The Judaica Press Complete Tanach with Rashi (trans. A. J. Rosenberg), via the Sefaria API — comment on Psalm 110:1 ('The word of the Lord to my master')
Patristic
Justin Martyr denies that Hezekiah is a priest forever after the order of Melchizedek or that he is Jerusalem’s redeemer. He also says Hezekiah did not send a rod of power into Jerusalem or rule in the midst of his enemies; rather, God averted his enemies after Hezekiah mourned and was afflicted.
Justin Martyr (c. 100-c. 165), Dialogue with Trypho (trans. Marcus Dods), in Ante-Nicene Fathers, vol. 1 (ed. Alexander Roberts and James Donaldson; Buffalo: Christian Literature Publishing Co., 1885) — ch. 83, 'It is proved that the Psalm, "The Lord said to My Lord," etc., does not suit Hezekiah'
Catholic
Aquinas says the Levitical priesthood was a figure of Christ's priesthood, not adequately representing it but falling far short, since it did not wash away sins and was not eternal, as Christ's is. The excellence of Christ's priesthood over the Levitical was foreshadowed in Melchisedech, who received tithes from Abraham, in whose loins the priesthood of the Law was tithed.
Thomas Aquinas (1225-1274), Summa Theologica, Part III (trans. Fathers of the English Dominican Province; New Advent edition) — Third Part, q. 22, a. 6: 'Whether the priesthood of Christ was according to the order of Melchisedech?'
Orthodox
Chrysostom says Jesus first questioned them, and because they did not answer the truth about Him—calling Him merely a man—He introduced David proclaiming His Godhead in order to overthrow their mistaken opinion.
John Chrysostom (c. 349-407), Homilies on the Gospel of St. Matthew, in Nicene and Post-Nicene Fathers, ser. 1, vol. 10 (ed. Philip Schaff; New York: Christian Literature Publishing Co., 1886) — Homily LXXI, on Matthew 22:34-46
Reformed / Protestant
Calvin argues the verse cannot be applied entirely and exclusively to David, as Christ's reply to the Pharisees shows. When they said Christ was to be David's son, Christ asked how David calls him Lord. Calvin calls the Jewish objection that Christ's reply was captious frivolous, because David speaks not in his own name but in that of the people.
John Calvin (1509-1564), Commentary on the Book of Psalms, vol. 4 (trans. James Anderson; Edinburgh: Calvin Translation Society, 1847) — commentary on Psalm 110:1-3
Evangelical
Wesley states that this dominion, to which David himself was subject, shows both the heavenly majesty of the king and the nature of his kingdom. To sit at my right hand means to remain in the highest authority and power.
John Wesley (1703-1791), Explanatory Notes upon the New Testament (1755), in Wesley's Notes on the Bible (CCEL) — note on Matthew 22:44