In verses 8-12, Paul quotes from Jeremiah 31:31-34. The writer begins by telling us that God found fault with the men of old, and this leads to the quotation from Jeremiah 31 in Hebrews 8:8.
From the failures of the past, Jeremiah turned his vision to the future. There are four significant things prophesied by Jeremiah and quoted by Paul about the new covenant in verses 10-12:
First, the New Covenant is inward and dynamic: It is written on the hearts and minds of the people. A shortcoming of the Old had been its outwardness. It had divinely given laws, but it was written on tablets of stone. Jeremiah looked for a time when people would not simply obey an external code but would be so transformed that God's own laws would be written in their inmost beings.
Second, there is a close relationship between the God who will be "their God" and the people, he says, who will be "My people." The change from the Old Covenant to the New Covenant is that while the formula of the covenant remains the same from age to age, it is capable of being filled with fresh meaning to a point where it can be described as a "new" covenant. "I will be your God" acquires fuller meaning with every further revelation of the character of God.
Third, all who enter it will have knowledge of God. There will be no need for a person to instruct his neighbor. The word rendered neighbor in verse 11 means "citizen," and thus a "fellow-citizen." Jeremiah moves from the wider relationship in the community to the narrower relationship in the family, saying that in neither case will there be a need to exhort anyone to know God because everyone will know Him.
This does not mean that under the conditions of the New Covenant there will be no place for a teacher. There will always be the need for those who have advanced in the Christian way to pass on to others the benefit of their knowledge. Rather, the meaning is that the knowledge of God will not be confined to a privileged few (as with the priesthood of ancient Israel). All those under the New Covenant will have their own intimate and personal knowledge of their God.
Fourth, under the New Covenant, sins are forgiven. Following repentance of sins and acceptance of the sacrifice of Jesus Christ, sins are forgiven. The superior sacrifice of Christ is offered once and for all, paying the penalty of sin for those who repent.
Martin G. Collins
The Law's Purpose and Intent