The writer continues proving that Jesus is greater than anyone. The author is writing to Jews that Jesus Is Greater Than Aaronic Priesthood.
My translation and notes of Hebrews 7:11-28
11 Therefore, if perfection was through the Levitical priesthood (for under it the people received the law) what further need [was there] that another priest should arise in the order of Melchizedek and not be mentioned in the order of Aaron?
12 Because the priesthood was being transferred, there needed to be a change in the law also.
[Israel has only one High Priest—the descendants of Aaron. For Jesus Christ to be High Priest shows the necessity of a new law and the fading of the old law (Hebrews 8:13). The old law is the Law of Moses (Joshua 8:31 and Acts 13:39) and the new law is the Law of Christ (Galatians 6:2). Jesus fulfilled Mosaic Law perfectly (all those appropriate for Jesus the Son of Man, since He is neither of Levi nor a woman].
13 For the one of whom these things are spoken about belongs to another tribe about which no one attended the altar.
[He speaks of Jesus, a descendant of Judah].
14 For [it is] obvious that our Lord arose from Judah, [and] about this tribe Moses said nothing pertaining to the priesthood.
15 Yet something is still more obvious―another priest arose, similar to Melchizedek,
16 Who was made [a priest] not according to the law about meeting human requirements but according to the power of an endless life.
17 For he testifies, “Yous are a priest forever in the order of Melchizedek.”
[Psalm 110:4. The Psalmist under inspiration declared that a new order of priesthood was coming, and thus, the canceling of the former laws concerning the High Priest was also coming].
18 So there truly is a canceling of the former commandment because of its weakness and ineffectiveness,
19 For the law made nothing perfect. But bringing in a better hopeN3 [did], by which we draw near to God.
[This is the reason the ‘former commandment,’ the Law of Moses, was weak and ineffective. The purpose of the Law was to convict people of their sins, not pay the penalty for their sins. It provided for a covering of the sins just like the first sacrifice provided Adam and Eve’s covering (Genesis 3:21). The Law reveals God’s righteous and holy nature (Leviticus 19:2), reveals the sinfulness of humans (Galatians 3:19), and Jesus Christ is the only savior providing forgiveness of our sins (Galatians 3:24 and Romans 10:4)]. Also see Isaiah 61:10 and 2 Corinthians 5:21].
20 None of this happened without an oath,
21 For indeed, they [Mosaic Law priests] were made priests without an oath, but this one [was made priest] with an oath by him who said to him, “The Lord swore and will not change his mind, ’Yous are a priest forever in the order of Melchizedek.’” [Psalm 110:4]
22 By so much more was Jesus made a guarantee of a better covenant.
23 Truly, there were many priests, because they were prevented from continuing by death,
24 But this one, because he endures forever, has a nontransferable priesthood.
25 Therefore, he is also able to save them, who come to God through him, to the uttermost, since he always lives to make intercession for them.
[The many superior traits of the High Priest—Jesus Christ. He lives forever, He is always and always will be High Priest, He is able to save (from God’s wrath), and always lives making intercession for those who place faith and trust into Jesus alone].
26 For such a high priest is suitable for us. [He is] holy, harmless, undefiled, separate from sinners, and made higher than the heavens,
[Holy in thoughts, desires, words, and deeds. Harmless since He does not extend revenge and punishment first but salvation. Undefiled as He is sinless. He is sanctified, separated, from all that is evil, and higher than heavens in His character traits, abilities, wisdom, love, etc. See 1 Corinthians 1:29-30].
27 Who doesn’t need daily, like those high priests, to offer up a sacrifice first for his own sins, then for the people’s. He did this [sacrifice for the people] once when he offered up himself.
28 The law appoints men high priests who have weakness, but the word of the oath, which came after the law, [appoints] the Son forever, who is completely perfect.
[Robertson (WORD PICTURES IN THE NEW TESTAMENT) writes:
Perfected (teteleiōmenon). Perfect passive participle of teleioō. The process (Hebrews 2:10) was now complete. Imperfect and sinful as we are we demand a permanent high priest who is sinless and perfectly equipped by divine appointment and human experience (Hebrews 2:17; Hebrews 5:1-10) to meet our needs, and with the perfect offering of himself as sacrifice.
Barclay (beware) writes (DAILY STUDY BIBLE):
The greatest of all the Levitical sacrifices began with a sacrifice for the sins of the High Priest. That was a sacrifice Jesus never needed to make, for he was without sin. The Levitical High Priest was a sinful man offering animal sacrifices for sinful people; Jesus was the sinless Son of God offering himself for the sin of all men. It was the law which had appointed the Levitical High Priest; it was the oath of God which gave Jesus his office; and because he was what he was, the sinless Son of God, he was equipped for his office as no human High Priest could ever be].