Among the Israelites a man became a priest by being born from the descendents of Levi who were of the family of Moses' brother Aaron, who was made a priest by God. A levitical priest ceased being a priest upon his death and the priesthood was passed on to his sons. In contrast was Melchizedek, a Gentile priest in the time of Abraham. The Bible is deliberately silent about his ancestry as a symbol that he did not inherit his priesthood.
Jesus is compared to Melchizedek to emphasize that his priesthood is different from the levitical priesthood. He does not inherit His priesthood but is made a priest directly by His heavenly Father. Moreover, His priesthood does not end with His death but continues because He is a priest forever. Incidentally the episode in today's gospel is just one instance whereby Jesus was showing that He supersedes the old order.
More significant for us is the fact that we do not worship alone in our eucharistic celebration. Jesus the priest is with us. He is present and active in you by virtue of baptism. He is present and active inthe inspired word of Scripture and in a pre-eminent way in the eucharist. Jesus as priest is our mediator. He brings the Father's truth and love to us, and He presents our prayers and sacrifice to the Father.
What we do at the eucharistic celebration has great meaning and value not because of ourselves but solely because of Jesus Christ.
Lord Jesus, You are the priest who draws me to honour God in my life.