The author of Hebrews points out that "perfection" was not attainable for us under the Levitical priesthood and system of law. The Greek word for "perfection" here means "completion, fulfillment and consummation". In the context of the book of Hebrews, this would refer to the completeness of our unity with Yahweh and receiving his rescue from our sin, as well as his promised rest and reward for his people.
Since this wasn't possible under the Levitical system, Yahweh had to raise up another, new kind of priest for us, "after the order of Melchizedek". This phrase refers to Psalm 110:4, a prophecy of the Messiah, the somewhat mysterious "Adonai"("Lord") often referred to in the Psalms who has both oneness with and distinction from Yahweh("LORD") in the Psalms.
A change in priesthood means a change in the law as well, since the Levitical priesthood was instituted by God's former laws for Israel. We'll get more details on that in chapter 8.
To solidify the fact that Jesus is a new kind of priest, the author observes that Jesus was not born from the tribe of Levi(where priests were normally selected from), but from Judah. Not only that but he has, in a sense, "genetic superiority" over the Levitical priesthood. Levitical priests were chosen because their bodies emerged from the Levite bloodline. Jesus is the ultimate high priest because his body is indestructible! This results in Jesus' ability to be a priest forever, as the author tells us with a direct quote of Psalm 110:4.
In this "change of priesthood", the former system is set aside. Not because it was "wrong" and a mistake is being corrected. It just wasn't up to the task of fixing our broken state and making us finally "complete" as we were meant to be. In that regard, the old system of law and sacrifices is weak and even useless.
Replacing it is a "better hope", that allows us to "draw near to God". That hope being Jesus, who not only redeems us so that we can escape our deserved judgment, but also journeys with us through the Holy Spirit, sympathizing with us in light of his own very real human experiences.
In talking with a few other geeks and looking at my own tendencies, I've discovered that a number of us have a hard time with Jesus being our "priest". Sure, we have no trouble trusting in him to perform the technical aspects of his priesthood, namely this once-for-all sacrifice which grants us the gift of eternal life with him. But the other part of his priesthood, that of "relational go-between" for us and God, is often ignored.
We might do other things that are good and result in a maturing understanding of scripture, like reading or studying the Bible or books by Christianity's greatest thinkers. We might involve ourselves in tasks, like giving or serving, that give us a sense of fulfillment as we encourage and bless others.
These are all good things, and we should keeping doing them and excelling at them. But knowledge can also "puff up"(1 Cor. 8:1) and tasks can keep us so busy that we don't really engage relationally with Jesus.
Specifically I mean prayer. Both scheduled time and random bursts throughout the day and night.
Last spring I began changing my "quiet time" routine to include lengthy (at least for me) stretches of silence and prayer. At first it felt stupid and unproductive. Or just plain boring. It takes a while for my mind to stop spinning with creative ideas and "to dos" and meditation or prayer both require considerable time for me to arrive at an undistracted state. The process sometimes involves anger, jotting down quick notes of random ideas to come back to later, more anger and frustration and frequently tears, as all my distractions, and self-justifications are stripped away, leaving me completely exposed (as I always truly am) before Jesus.
In those times, which often represent only a small fraction of the entire time I devote to the experience, I am faced and defeated by my shortcomings and selfishness. But then I'm also immediately rescued by the unrelenting love of Jesus. I come away from those times feeling lighter than before, with the formerly heavy stresses of the day put in perspective. After seeing my own hidden sin exposed in the presence of Jesus, in light of his forgiveness, I'm able to better discern the difference between what my agenda is and what God's agenda is, and shift my course accordingly.
I think this aspect of Jesus' priesthood is part of the rest and reward the author of Hebrews encourages us to pursue: Resting in the love and forgiveness of Jesus and, as a result, being better able to invest in his plans for the world and for each of us.