The nature of the new covenant over against the old begins to be elaborated in verse 10. In Jeremiah's context, establishing a new covenant would solve the problem of Israel's continual violation of covenant stipulations. The promises made to Abraham can hardly become a reality if Israel continues to transgress what God commands. The Lord declares, however, that he will remedy the problem, promising to make a new agreement, a new covenant, with Israel. The author Hebrews believes the new covenant has been fulfilled in his day, in the cross and resurrection of Jesus Christ.
The new covenant is distinct from the old in that God inscribes his law on their hearts and minds. Under the old covenant Israel knew the law as a written statue external to them, but knowing the commands did not give them any inclination to keep what God commanded. Jeremiah complained that they had an uncircumcised heart (Jer. 4:4; 9:25-26). The new covenant is an entirely different state of affairs, for now God circumcises the heart of his people in accord with the promises of Deut. 30:6 ("The LORD your God will circumcise your heart and the hearts of your descendants, and you will love Him with all your heart and all of your soul so that you will live"). God's commands are not an onerous burden but reflect the desires of their heart since God has imprinted the law upon their hearts. The fundamental flaw in God's people has been remedied, for now they delight to do the will of God.
The covenant formula would, under the terms of the new covenant, become a reality. God would be their God, and Israel would be his people (cf. Gen. 17:8; Exod. 6:7; Jer. 30:22; 32:28; Ezek 11:20; 36:28; 37:23,27). God would meet every need of his people, protecting and guarding them from all that would harm them. At the same time Israel would be the people of God. In the NT, this promise is fulfilled in the church of Jesus Christ, which is understood to be the new Israel." pg. 251,252 "Commentary On Hebrews" by Thomas R. Schreiner (Biblical Theology For Christian Proclamation)
" For if that first covenant had been faultless, then should no place have been sought for the second.
 For finding fault with them, he saith, Behold, the days come, saith the Lord, when I will make a new covenant with the house of Israel and with the house of Judah:
 Not according to the covenant that I made with their fathers in the day when I took them by the hand to lead them out of the land of Egypt; because they continued not in my covenant, and I regarded them not, saith the Lord.
 For this is the covenant that I will make with the house of Israel after those days, saith the Lord; I will put my laws into their mind, and write them in their hearts: and I will be to them a God, and they shall be to me a people:
 And they shall not teach every man his neighbour, and every man his brother, saying, Know the Lord: for all shall know me, from the least to the greatest.
 For I will be merciful to their unrighteousness, and their sins and their iniquities will I remember no more.
 In that he saith, A new covenant, he hath made the first old. Now that which decayeth and waxeth old is ready to vanish away." Hebrews 8:7-13