The name means "my King is righteousness".
See some of Spurgeon's exposition on Psalm 110 (The Treasury of David)
Verse 4. We have now reached the heart of the psalm, which is also the very centre and soul of our faith. Our Lord Jesus is a Priest King by the ancient oath of Jehovah: "he glorified not himself to be made an high priest," but was ordained there unto from of old, and was called of God an high priest after the order of Melchizedek. It must be a solemn and a sure matter which leads the Eternal to swear, and with him an oath fixes and settles the decree for ever; but in this case, as if to make assurance a thousand times sure, it is added," and will mot repent." It is done, and done for ever and ever; Jesus is sworn in to be the priest of his people, and he must abide so even to the end, because his commission is sealed by the unchanging oath of the immutable Jehovah. If his priesthood could be revoked, and his authority removed, it would be the end of all hope and life for the people whom he loves; but this sure rock is the basis of our security -- the oath of God establishes our glorious Lord both in his priesthood and in his throne. It is the Lord who has constituted him a priest for ever, he has done it by oath, that oath is without repentance, is taking effect now, and will stand throughout all ages: hence our security in him is placed beyond all question.
The declaration runs in the present tense as being the only time with the Lord, and comprehending all other times. "Thou art," i.e., thou wast and art and art to come, in all ages a priestly King. The order of Melchizedek's priesthood was the most ancient and primitive, the most free from ritual and ceremony, the most natural and simple, and at the same time the most honourable. That ancient patriarch was the father of his people, and at the same time ruled and taught them; he swayed both the sceptre and the censer, reigned in righteousness, and offered sacrifice before the Lord. There has never arisen another like to him since his days, for whenever the kings of Judah attempted to seize the sacerdotal office they were driven back to their confusion: God would have no king priest save his son. Melchizedek's office was exceptional none preceded or succeeded him; he comes upon the page of history mysteriously; no pedigree is given, no date of birth, or mention of death; he blesses Abraham, receives tithe and vanishes from the scene amid honours which show that he was greater than the founder of the chosen nation. He is seen but once, and that once suffices. Aaron and his seed came and went; their imperfect sacrifice continued for many generations, because it had no finality in it, and could never make the comers thereunto perfect. Our Lord Jesus, like Melchizedek, stands forth before us as a priest of divine ordaining; not made a priest by fleshly birth, as the sons of Aaron: he mentions neither father, mother, nor descent, as his right to the sacred office; he stands upon his personal merits, by himself alone; as no man came before him in his work, so none can follow after; his order begins and ends in his own person, and in himself it is eternal, "having neither beginning of days nor end of years The King Priest has been here and left his blessing upon the believing, and now he sits in glory in his complete character, atoning for us by the merit of his blood, and exercising all power on our behalf."
"O may we ever hear thy voice
In mercy to us speak,
And in our Priest we will rejoice,
Thou great Melchizedek."
The last verses of this psalm we understand to refer to the future victories of the Priest King. He shall not forever sit in waiting posture, but shall come into the fight to end the weary war by his own victorious presence. He will lead the final charge in person; his own right hand and his holy arm shall get unto him the victory.