Just what do you mean "A New Covenant"?
God made a promise in Old Testament times through Jeremiah the prophet that he would bring in a NEW Covenant which the people would not break, unlike the Old Covenant which was in existence at that time.
Jer 31:31-34 "Behold, the days are coming, declares the LORD, when I will make a new covenant with the house of Israel and the house of Judah, not like the covenant that I made with their fathers on the day when I took them by the hand to bring them out of the land of Egypt, my covenant that they broke, though I was their husband, declares the LORD. But this is the covenant that I will make with the house of Israel after those days, declares the LORD: I will put my law within them, and I will write it on their hearts. And I will be their God, and they shall be my people. And no longer shall each one teach his neighbor and each his brother, saying, 'Know the LORD,' for they shall all know me, from the least of them to the greatest, declares the LORD. For I will forgive their iniquity, and I will remember their sin no more."
The writer of Hebrews (Paul?) confirms that God has kept this promise through Jesus, the Christ, by repeating it in the context of it being fulfilled, in Chapter 8 verses 8-12.
Heb 8:8-12 For he finds fault with them when he says: "Behold, the days are coming, declares the Lord, when I will establish a new covenant with the house of Israel and with the house of Judah, not like the covenant that I made with their fathers on the day when I took them by the hand to bring them out of the land of Egypt. For they did not continue in my covenant, and so I showed no concern for them, declares the Lord. For this is the covenant that I will make with the house of Israel after those days, declares the Lord: I will put my laws into their minds, and write them on their hearts, and I will be their God, and they shall be my people. And they shall not teach, each one his neighbor and each one his brother, saying, 'Know the Lord,' for they shall all know me, from the least of them to the greatest. For I will be merciful toward their iniquities, and I will remember their sins no more."
This sets the scene for the writer to let us know that with the NEW Covenant coming into force the Old Covenant, made at Sinai, becomes obsolete.
Heb 8:13 In speaking of a new covenant, he makes the first one obsolete. And what is becoming obsolete and growing old is ready to vanish away.
When you read God's promise as given through Jeremiah and repeated by Paul, you could not have failed to notice that there were a number of parts to the promise.
Paul confirms that the first part of the promise (that the Law of the NEW Covenant would be internal to the people and not an external code written on stone), has been completed, by what he writes in 2nd Corrintians chapter 3, verses 3-6...
2Co 3:3-6 And you show that you are a letter from Christ delivered by us, written not with ink but with the Spirit of the living God, not on tablets of stone but on tablets of human hearts. Such is the confidence that we have through Christ toward God. Not that we are sufficient in ourselves to claim anything as coming from us, but our sufficiency is from God, who has made us competent to be ministers of a new covenant, not of the letter but of the Spirit. For the letter kills, but the Spirit gives life.
Fulfilling the second part of the promise to remember our sins no more (for those who partake of the NEW Covenant), God sent Jesus to die for us; which he did, willingly on the cross. We can see this inaugurated in the book of Luke, chapter 22, verses 19-20. Our forgiveness was bought at the price of Jesus Christ's death.
Luk 22:19-20 And he took bread, and when he had given thanks, he broke it and gave it to them, saying, "This is my body, which is given for you. Do this in remembrance of me." And likewise the cup after they had eaten, saying, "This cup that is poured out for you is the new covenant in my blood.
And Paul repeats this point in the 1st book of Corrinthians, chaper 11, verses 23-26.
1Co 11:23 For I received from the Lord what I also delivered to you, that the Lord Jesus on the night when he was betrayed took bread, and when he had given thanks, he broke it, and said, "This is my body which is for you. Do this in remembrance of me." In the same way also he took the cup, after supper, saying, "This cup is the new covenant in my blood. Do this, as often as you drink it, in remembrance of me." For as often as you eat this bread and drink the cup, you proclaim the Lord's death until he comes.
And the writer of Hebrews also confirms this was done for us by Jesus in Hebrews, chaper 9, verses 11-15.
Heb 9:11-15 But when Christ appeared as a high priest of the good things that have come, then through the greater and more perfect tent (not made with hands, that is, not of this creation) he entered once for all into the holy places, not by means of the blood of goats and calves but by means of his own blood, thus securing an eternal redemption. For if the sprinkling of defiled persons with the blood of goats and bulls and with the ashes of a heifer sanctifies for the purification of the flesh, how much more will the blood of Christ, who through the eternal Spirit offered himself without blemish to God, purify our conscience from dead works to serve the living God. Therefore he is the mediator of a new covenant, so that those who are called may receive the promised eternal inheritance, since a death has occurred that redeems them from the transgressions committed under the first covenant.
Now the thing to ask is, "What was it that was placed into those who partake of the NEW Covenant in order for God to put his Law into our hearts?Jesus himself told us what that would be...
Joh 16:12-13 "I still have many things to say to you, but you cannot bear them now. When the Spirit of truth comes, he will guide you into all the truth, for he will not speak on his own authority, but whatever he hears he will speak, and he will declare to you the things that are to come.
It's the Holy Spirit within us who is our Guide and Helper and who is the Law we follow. Through him we have Jesus Christ in us, the hope of glory.
So summing up;
The other main promise is that God would forgive our sins and remember them no more. Under the Old Covenant forgiveness came via the blood of bulls and goats which could not take away sins.
We though have forgiveness through the blood of Jesus Christ, by his death on the cross, giving us eternal redemption when we accept his sacrifice in our stead.
Now, having accepted the blood of Jesus as atonement for our sin, we are reconciled to God. He is our God and Lord and we are truly his people.
This Covenant and it's promises are unlike the previous covenant and in fact makes the Old (Sinai) Covenant obsolete.
This is the NEW Covenant that gives Christians LIFE!