The words used for "lord" in Psalm 110 are different. The first word is "Yahweh" and refers to the God of Abraham, Isaac and Joseph. The second word is "Adonai" and refers to Christ. In Matt. 22:44, Jesus asks about the identity of the Messiah in relationship to David and the Pharisees reply that he is the Son of David. In fact, Son of David was a common name for the Messiah in Christ's day.
So Jesus is trying to make them see that, if the Son of David is the Messiah, and David calls him "Adonai", then he must be more than a physical descendant of David. According to Psalm 110, this "Adonai", the Messiah, was alive at the same time that David was and it's obvious that he was greater than David because David called him "Adonai".
The Pharisees don't answer because Jesus has set it up so that the only sensible and intelligent answer is one that they don't want to give. Just prior to this, they tried to trick him with a question about taxes. Then the Sadducees had tried to trick Jesus with a question about multiple wives and life after death -- a particularly silly question coming from the Sadducees as they didn't even believe in life after death. So the Pharisees, hearing about the failure of the Sadducees to trick Jesus, tried to do so again themselves with a question about the greatest commandment. And then it was Jesus' turn to confront them with a question that shut them up.
Note that Psalm 110 is talking about Jesus as "Adonai" because it describes him in the following ways:
� He sits at God’s right hand (verse 1)
� He will triumph over all His enemies and rule over them (verses 1–2)
â—� He will lead a glorious procession of troops (verse 3)
� He will be “a priest forever, in the order of Melchizedek� (verse 4)
� He will have divine power to crush kings, judge nations, and slay the wicked (verses 5–6)
â—� He will find refreshment and be exalted (verse 7)
These things all refer to Christ and are borne out in other Scripture passages in both the New Testament and the Old. Take, for example, the fact that he sits at God's right hand. See here for the significance of that:
https://www.gotquestions.org/right-hand-God.html
Good question, Checkpoint. Thanks for asking it.