“The days are coming,” declares the LORD,
“when I will make a new covenant with the people of Israel
and with the people of Judah.
It will not be like the covenant I made with their ancestors
when I took them by the hand to lead them out of Egypt,
because they broke my covenant, though I was a husband to them declares the LORD.
“This is the covenant I will make with the people of Israel
after that time,” declares the LORD.
“I will put my law in their minds and write it on their hearts.
I will be their God, and they will be my people.
No longer will they teach their neighbor, or say to one another, ‘Know the LORD,’
because they will all know me,
from the least of them to the greatest,”
declares the LORD.
“For I will forgive their wickedness
and will remember their sins no more.”
God spoke through the prophet Jeremiah of a New Covenant that He would make with His people. It would be a new and better covenant. Whereas with the Old Covenant, God gave His people the law. Yet, they were never able to obey. They're hearts were sinful. They needed a heart transplant. With the Old Covenant, sin caused a separation of intimacy between God and His people. Though God was their husband, though He brought them out of Egypt, and gave them a King (David), and the prophets, Israel repeatedly turned from knowing God in an intimate way, as we were all created to know him. With the Old Covenant, God's people had to constantly atone for their sins through repeated sacrifices. They needed animal sacrifices and the work of the high priests for forgiveness.
However, the in the New Covenant, God fulfilled these things in Jesus. Through the New Covenant, we have a new experience of obedience, a new experience of intimacy with God, and a new experience of forgiveness. The Son came from heaven to earth, taking on human flesh, died on the cross, and resurrected in order to give us this new experience; in order that He would be our God, and we would be His people.
In the New Covenant, we are united to Christ, and are therefore indwelled with the Holy Spirit. The Spirit gives us a new heart and new life. In our old life, we loved sin. We lived according to the flesh, according to the world. Our taste buds were for sin. Our delight was in sin. Yet, through Jesus, the Spirit dwells in us. The Spirit gives us new hearts; hearts that desire and long to obey God. The Spirit gives us new taste buds. Whereas in the past, sin tasted so good to our souls, it now is disgusting. Our taste buds long to feast on the things of God. We live according to the Spirit. God's law is no longer a burden, but it is a delight. It has been written on our hearts. (Romans 8)
In the New Covenant, we are united to Christ and therefore have an deep intimacy with God. Whereas before, sin and death inhibited us from having fellowship with God. Now, Jesus is our covenant representative. We have access to God through Christ. We were once enemies, but have been made friends. We were once fatherless, but are now called Sons/Daughters! The natural intimacy that Adam and Eve had with God before they rebelled can now be experienced. Jesus said, "I am the good shepherd. I know my own and my own know me" (John 10:14), and "If you had known me, you would have known my Father also. From now on you do know him and have seen him." (John 14:7)
Lastly, in the New Covenant, we have a new experience of forgiveness. This forgiveness is different from the Old Covenant in that it is COMPLETE. It is complete in that this sacrifice was once for all. It was for ALL the sins of His people. No longer would God's people have to look to repeated sacrifices of atonement for forgiveness. Rather, with the New Covenant, they would look to their New Covenant representative for forgiveness. The Son came to earth, and died as the all time sacrifice for sins. Our sin greatly angers God. Our sin deserves wrath. It deserves punishment. That is why in the Old Testament, people had to constantly atone for their sins. Something had to pay the price. God is just. Sin can't go unnoticed. But the gospel tells us that God has made a new Covenant, where He would do the most radical thing. His Son would come to earth, and take the once for all punishment for our sins. Even though we have been unfaithful from the beginning. God so loved the world that He (the Son) sacrificed himself on the Cross for His people's sins, once and for all. Let's look at a few verses from Hebrews and Romans...
"he (Jesus) entered once for all into the holy places, not by means of the blood of goats or calves, but by means of his own blood, thus securing an eternal redemption." Hebrews 9:12
"But as it is, he has appeared once for all at the end of the ages to put away sin by the sacrifice of himself." Hebrews 9:27
"For all have sinned and fall short of the glory of God, and are justified by his grace as a gift, through the redemption that is in Christ Jesus, whom God put forward as a propitiation by his blood, to be received by faith" Romans 3:23-25
"having forgiven us all our trespasses, by canceling the record of debt that stood against us with its legal demands. This he set aside, nailing it to the cross." Colossians 2:13-14