Hebrews 8:6-8 meaning

The Pauline Author explains what Jesus has accomplished for us: But now He has obtained a more excellent ministry, by as much as He is also the mediator of a better covenant, which has been enacted on better promises (v 6). Covenant just means "agreement" or "contract." 

A typical human contract has two parties, where each party agrees to do something for the other party. In fact, in western law, a contract is not valid unless there is "consideration"—meaning a contract is not valid unless each side is getting someone of value. God's covenants are both similar as well as vastly different. His covenant is different because God makes some promises unilaterally—we don't have to do anything for Him to receive the benefit (in the instance of Jesus's sacrifice atoning for our sins, we are not capable of doing anything anywhere near an equivalent value). 

God is our inheritance no matter what we do, simply because we believe. But some of God's promises do require us to do something, as in the case of the reward of the inheritance. That reward requires that we be faithful.

The New Covenant has better promises on both counts, both conditional as well as unconditional blessings.

The earthly priests, the Levites, served the Old Covenant, which is the Law of the Old Testament. God chose Israel as His people and included in His law an open door to accept any "alien or foreigner" who desired to come under the covering of Israel. In fact, the Bible highlights the stories of Ruth and Rahab as two foreign women who believed in the God of Israel and were rewarded for their faith.

Deuteronomy 30:16-18 summarizes the conditional part of the Old Covenant: 

"In that I command you today to love the Lord your God, to walk in His ways and to keep His commandments and His statutes and His judgments, that you may live and multiply, and that the Lord your God may bless you in the land where you are entering to possess it. But if your heart turns away and you will not obey, but are drawn away and worship other gods and serve them, I declare to you today that you shall surely perish."

The Pauline Author's point here is that if that first covenant had been faultless, there would have been no occasion sought for a second (v 7). There would be no need to establish a new covenant if the first one was sufficient. The law had sacrifices that atoned for sin, but these atonements were not permanent. And the law directed the Israelites in how to live, but it did not change their hearts. 

The bulk of the Old Testament is about Israel's consistent rebellion toward God, no matter how many times He showed them protection and mercy. In this passage, the Pauline Author quotes Jeremiah 31:31-34, where God declares that He will one day create a new covenant with Israel and Judah (the Israelites): For finding fault with them, He says, "Behold, days are coming, says the Lord, when I will effect a new covenant with the house of Israel and with the house of Judah" (v 8).

Jesus is both the leader and the agent of this New Covenant; He is the person who goes between us and God. His death is what atoned for our sins forever, and His blood is what sanctifies us (Hebrews 13:12). The key difference between new and old is centered in the human heart. Jesus's sacrifice atoned for sins once and for all, and cleared every bit of guilt in the presence of God, for all who believe. 

Then Jesus's resurrection and ascent to heaven paved the way for Him to send the Holy Spirit to inhabit the hearts of those who believe. This is what the Pauline Author refers to as a great mystery, the fact that Jesus is now within us.

Because Jesus became a man and suffered death on the cross, He has been rewarded with authority to reign over the earth (Matthew 28:18, Philippians 2:5-9, Hebrews 2:9). Jesus wants to share this reward with all believers who are faithful servants; these are the "many sons" He hopes to bring to share His glory, if they endure rejection from the world and live by following the Spirit's leading in their lives (Hebrews 2:10; Revelation 3:21). 

The New Covenant, which has brought Jesus and the Spirit to live within us and guide us, makes a way for us to live a life of faithful obedience to God for which He wants to reward us at the Judgement. 

The Pauline Author will continue to quote Jeremiah to show the contrast between the Old Covenant of the law, and the New Covenant of grace through Jesus Christ.