HEBREWS 8:6 & 9:15:
Again the question is raised. If the old covenant could have achieved God's purposes, then there would have been no need of a better covenant. (vs7)Christs ministry is far more excellent than the high priest for what he has to offer, which the High Priest could not is to mediate the new covenant. While it is clear that men have always in the matter of forgiveness of sins, had to come through the priests as mediators for them. So where the high Priests continually offered sacrifices that couldn't perfect the consciences of the worshipper. The ministry of Jesus does perfect the consciences of the worshipper. That means the promises of forgiveness weren't as good as the new covenant promises of forgiveness. The Old Covenant promises only made the worshipper ceremonially "clean". The new promises of forgiveness through the sacrifice of Christ make a person inwardly clean. it deals with hardness of heart in an effective and dynamic way. So his ministry is better because it is based on better promises, than the Old covfenant.
Jesus doesn't only offer sacrifice for his people's sins.He also mediates the covenant. So again in this sense is greater than any other High Priest. HEBREWS 9:15-16
The sense that it carries is that he must mediate the promises made in the New covenant. here the equivelant would be a solicitor or lawyer who is called in when a last will and testament must be settled. A last will and testament is quite similar to the New covenant. It contains promises to those who are in the will, to inherit what is promised. The word "diatheke" is used for new covenant where the old covenant is called "suntheke". Diatheke is the word for a last will and testament. it is a covenant of promise. The death of the one who made the promises must be established for the wills promises to be ratified. To ratify the covenant means to bring the promises of the last will and testament to be brought into effect. The way this is done is through a mediator. The lawyer's (mediator) task is to make sure that the will is actually the will of the person named. He then makes sure that what is promised, actually goes to whom it is promised. So Jesus by his death ratifies ie brings into effect the promises made to God's elect. But more than that he makes sure the persons mentioned in the will,actually inherit what is promised. He dispenses the promises. So how does that effect believers in Jesus Christ ? When a person becomes one mentioned in the will through faith in Jesus. Jesus makes sure the promises to them are made effectual in their lives. He makes sure that the promise of the Spirit, is given them. Thereby he ensures that the person will live in Godliness by observing God's laws and statuetes. He makes sure the promises of blanket forgiveness and the promises of a new heart and mind are made effective in the lives of those who inherit these promises. Who is the one who inherits the promises of the new Covenant? it is the person who draws near to God through trusting in Jesus Christ to save them. Those who inherit are the believers in God's son and His promises. For these true believers salvaton is eternally secured.

So Christ is greater than the old Testament High Priests for he has offered a once and for all sacrifice. The old Priests had to continually offer sacrifice. One had to offer sacrifice after sacrifice, because the sacrifice would only be effective for his forgiveness until he sinned again. Then he would have to offer sacrifice sacrifice again and again. Not so who worship God by the once and for all sacrifice of Christ. Christ's sacrifice covers past sins,present sins,and even future sins. If this is not the case then His sacrifice is not once and for all. Immediatly I hear the objection!! That this teaching gives us licence to sin. Not at all, for those who are partakers of this blanket amnesty are New creations. The new covenant implies this new creation. For in Jeremaiah31 & Ezekiel 36 It is the recreated people who inherit the recreated land(The Kingdom of God) What I am saying is in the light of what has already been said. If Christ makes the promises effective in you. Then you are a New Creation. That means that your being has been transformed. It means you have been changed from the inside out. You are now a person who wants to do what pleases God. Your desires are now inclined toward righteousness. You also have the fear of God in you so that you will not depart from God. If you are a partaker of the New Covenant it means you have partaken of the 1st fruits of the New Creation. We are in the kingdom, but at the same time we are not yet in the Kingdom. This means that even though we are delivered from sin now, we will only be sinless at the establishment of God's Kingdom. We have by the power of Christ begun to remove the taint of sin until we are sinless. This is one of the manifestations whereby Jesus has begun to remove the curse, from us and creation. This is what it means to be Born again. Born again is new Creation language. If the work of recreation has begun by the new covenant promises being made effective in you. You can be sure that Christ will continue that work in you until the day of Christ Jesus. The day of Christ jesus is at His return when he makes all things new.

The following will explain the way the new covenant is exactly the same as a last will and testament. The legal requirements are that there is a "TESTATOR" the one who wishes to make the testament or covenant. In this case it would be YAHWEH who is God. Jesus had to die for the covenant or testament to come into effect. It is a covenant where those named will inherit. Usually it is the sons who inherit. So those who inherit the promises are the sons of God. Then there is the mediator who must execute the will of the deceased.This would usually be a lawyer. So let's look at the document.