Hebrews 8:9-13 meaning

In Hebrews 8:8-12, the Pauline Author quotes from Jeremiah 31:31-34. The quote continues on from the immediately preceding text: 

Not like the covenant which I made with their fathers
on the day when I took them by the hand
to lead them out of the land of Egypt;
for they did not continue in My covenant,
and I did not care for them, says the Lord. 

"For this is the covenant that I will make with the house of Israel
after those days, says the Lord:
I will put My laws into their minds,
and I will write them on their hearts.
And I will be their God,
and they shall be My people.
"And they shall not teach everyone his fellow citizen,
and everyone his brother, saying, 'Know the Lord,' for all will know Me,
from the least to the greatest of them.
"For I will be merciful to their iniquities,
and I will remember their sins no more." 

The Pauline Author is quoting from this passage to show that God told the Israelites of the Old Testament that one day He would make a new covenant with them, a better covenant. This emphasizes to the Jewish Christian readers of this letter that this message is not new — it was promised to Israel long before (at the time of the writing of this letter, it would have been roughly 500 years after the writing of Jeremiah).

The ministry of our High Priest, Jesus, far surpasses the ministry of the priests who served under the Old Covenant. The prior ministry of the priests was to act as a mediator between God and His people by continually offering sacrifices to atone for sins and offerings of peace and reminding the people of the law of God. 

Jesus's sacrifice brought about the New Covenant, where the law was no longer just written on stone tablets but is now written on the hearts of all who believe. The location of the law had to change. Not only that, but there was also now a permanent sacrifice that atoned for the sins of the world, once and for all.

The Old Covenant did not work. Writing the law on tablets and counting on earthly priests to mediate between God and His people did not and could not bring about the result of God and man at peace. Israel rebelled against God time and again, turning to idolatry, injustice, and immorality. God's plan to provide a full pardon for the sins of mankind and to offer salvation from sin and death was dependent upon a better covenant.

The Pauline Author makes this point in Hebrews 7:11, "Now if perfection was through the Levitical priesthood (for on the basis of it the people received the Law), what further need was there for another priest to arise according to the order of Melchizedek, and not be designated according to the order of Aaron?"

His New Covenant, utilizing a blameless sacrifice in order to atone, once and for all, for the sins of mankind, was established. The sacrifice was a perfect, sinless man, fully obedient to the will of God; and He became a Son, our King, and our Priest.

This New Covenant will be between the houses of Israel and Judah and God, with the tribe of Judah producing the blood sacrifice of that covenant. Jesus Himself, who was born of the tribe of Judah, stated at the Last Supper, "for this is My blood of the covenant, which is poured out for many for forgiveness of sins" (Matthew 26:28). 

This covenant is also extended to the nations, meaning the Gentiles, meaning all people on the earth, even those who aren't Jewish. In Romans 9:25-26, Paul cites the prophet Hosea to establish this acceptance of Gentiles, 

"As He says also in Hosea, 'I will call those who were not My people, 'My people,' And her who was not beloved, 'beloved.' 'And it shall be that in the place where it was said to them, 'you are not My people,' There they shall be called sons of the living God.'"

This New Covenant would be far greater than the one that the Lord established with the Israelites when He took them by the hand to lead them out of Egypt and out of slavery. He would write His law and commands on the hearts and minds of His people, bind it to their very being, so that they have it with them always. In these days, we do not have to go through earthly priests and teachers to know the Lord, for we will all know Him and have the ability to approach Him boldly, from the least to the greatest. He will have mercy on us and grant us a full pardon for our sins (v 12). 

With this covenant, we have a clear view of God, and on a practical level, we will be better able to determine that which is holy from that which is profane, because the law has been written into our hearts. After initial faith in Christ, we have been given new hearts (Jeremiah 24:7), and we are indwelled by the Holy Spirit, who is our Helper (John 14:26).

Our new hearts and our Helper, the Holy Spirit, make it possible for us to be faithful to God in all of our actions, if we obey the Spirit's leading, which is the path to gaining the reward of our inheritance (Colossians 3:23). The reward for faithful believers is to be "crowned with glory and honor" (Hebrews 2:7-9), to be reinstated to the job we were originally created for, to reign over the earth in harmony with God, one another, and creation. This is what Jesus, in establishing the New Covenant, is trying to lead us to (Hebrews 10). We are to live a life trusting that God's ways are for our best, just as Jesus did (Philippians 2:5-9).

We live under this New Covenant now, thanks to the death and resurrection of Jesus Christ. Due to His atoning work on our behalf, by faith in Him, we are now able to please God. The author writes, When He said, "A new covenant," He has made the first obsolete (v 13). The previous covenant is becoming obsolete and growing old and is now ready to disappear (v 13).