Christ Is The Superior High Priest.
Christ’s priesthood is after the order of Melchizedek. Hebrews 7:1-3.   
In this chapter, the author shows that the priesthood of Christ is superior to the Levitical priesthood because it is after the order of Melchizedek. The Old Testament priests came from the line of Levitical priesthood (Hebrews 7:11).  It was impossible for Jesus to  become a priest because He was from the tribe of Judah. Melchizedek is an Old Testament type of Christ. Only twice Melchizedek is mentioned in the Old Testament (Genesis 14:18-20; Psalms 110:4). In chapter 7:1-3, we see the historical account of Melchizedek.  His name means king of righteousness. He was the king of Jerusalem. So he was a king of righteousness and peace. He was also the priest of the most high God (Genesis 14:18). He was without father, without mother, without descent, having neither beginning nor end, but made like unto the Son of God (Hebrew 7:3). There is not any record of his birth or death in the Scripture.  He resembles the Son of God who has no beginning or end. 

The priesthood of Melchizedek is superior to Levitical Priesthood. Hebrews 7:4-10.
Melchizedek resembles Jesus in several ways. He blesses the great patriarch Abraham, implying that he is superior to Abraham.  Abraham, the ancestor of Levite, paid tithe to Melchizedek. When Abraham paid tithe to Melchizedek, Levi was in turn offering tithe to Melchizedek.  Levi was descended from Abraham. Without any dispute the lesser is blessed by the better (Hebrews 7: 7, 8). So Melchizedek  was superior to Israel's most revered ancestor. Old Testament priests could not give people complete removal of sin and reconciliation to God. Law and the Levitical priesthood are closely connected.  So it became necessary to have another order of priesthood to forgive sins and bring reconciliation to God (Hebrews 7:11, 12). Since Jesus' priesthood is associated with Melchizedek, He is superior to Aaron. 

Christ is superior, because He lives forever to make intercession. Hebrews 7:11- 17.  
Jesus' priesthood is after the order of Melchizedek. Aaron was from the tribe of Levi but Christ came from Judah. Aaron's priesthood was installed by legal procedures (Hebrews 7:16).  But Christ's priesthood was confirmed by God's oath. "You are a priest forever, after the order of Melchizedek" (Hebrews 7:17; Psalms.110:4). Levitical priesthood  was changeable because of death. When Aaron died, he was replaced by his son. So Christ became High Priest, not by meeting the requirements of law but by the power of God (Hebrews 7:16). Christ's priesthood is eternal because He lives forever. The Mosaic Law was temporary and made nothing perfect, including the Old Testament priesthood. It could never fully remove our sins or enable men to draw near to God. Whereas Christ gives us hope because He lives forever to make intercession for us.  Jesus is the ultimate and final priest who represents us before God.  He is the only go-between (1 Timothy 2:5). Christians should rejoice that they can now come directly to the Father through Jesus Christ.

Christ has become the surety of a better covenant.  Hebrews 7:18-28. 
Jesus' priesthood gives the guarantee of a better covenant. This new covenant is eternal and established by Christ. It gives forgiveness of sin to all those who come to God through Christ (7:20-22).  Christ's priesthood is permanent because He lives forever. There were many high priests under the old covenant. Death prevented them from continuing in their office (Hebrews 7:23). But Christ stands alone and His priesthood will never end. Therefore He is able to completely save everyone who comes to God through Him (7:25). Also Christ is our perfect sacrifice and ideal High priest.  Being perfect, He did not need to offer daily sacrifice for His own sins (Leviticus 4:2, 3). But Christ offered Himself for our sins on the cross.  His sacrifice was a willing and conscious sacrifice unlike animal sacrifice. Old Testament sacrifices were sacrifices of unwilling and unthinking animals. Christ’s sacrifice is the once for all sacrifice and never to be repeated. For He is holy, blameless, unstained, separated from sinners and exalted above the heavens (Hebrews 7:26).  He is both our perfect sacrifice and our great High Priest.  He lives forever to make intercession for us.  Do you know Jesus as your Lord and Savior?