The new covenant is a significant concept in the Bible, primarily prophesied in the Old Testament and believed by Christians to be established by Jesus Christ in the New Testament. The most well-known prophecy of this covenant is in the book of Jeremiah 31:31-34, which states:

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"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. For 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.”

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Key aspects of this new covenant, as described in Jeremiah, include:

 * Internalization of the Law: Unlike the Mosaic Covenant, which was written on tablets of stone, the new covenant involves God placing His law directly "in their minds" and "on their hearts." This signifies an inward, heartfelt desire to obey God, rather than a forced, external set of rules.

 * Forgiveness of Sins: A central promise of the new covenant is the complete forgiveness of sins, with God vowing to "remember their sin no more."

 * A Personal Relationship with God: The prophecy indicates a time when everyone, from the least to the greatest, will "know the Lord," implying a deep and personal relationship with God that doesn't require an intermediary to teach them.

In Christianity, the New Testament explains that Jesus' death and resurrection fulfilled this prophecy, establishing the new covenant through his blood (Luke 22:20). This covenant is seen as a fulfillment of God's promises to Israel, but also as something that extends to all who believe, both Jewish and Gentile.

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The New Testament explains that Christians—both Jewish and Gentile (non-Jewish) believers—are included in the new covenant. This is a central theme in several New Testament books, especially those written by the Apostle Paul.

Here are some key passages that explain this:

1. Ephesians 2:11-22

This passage is one of the clearest and most direct explanations. Paul tells the Gentile Christians in Ephesus to remember their past state:

> "Therefore, remember that formerly you who are Gentiles by birth and called 'uncircumcised' by those who call themselves 'the circumcision' (which is done in the body by human hands)—remember that at that time you were separate from Christ, excluded from citizenship in Israel and foreigners to the covenants of the promise, without hope and without God in the world." (Ephesians 2:11-12)

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He then explains that through Jesus's sacrifice, this has all changed:

> "But now in Christ Jesus you who once were far away have been brought near by the blood of Christ. For he himself is our peace, who has made the two groups one and has destroyed the barrier, the dividing wall of hostility... His purpose was to create in himself one new humanity out of the two, thus making peace, and in one body to reconcile both of them to God through the cross..." (Ephesians 2:13-16)

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This passage shows that Gentiles, who were once "foreigners to the covenants," are now "fellow citizens with God's people and also members of his household" (Ephesians 2:19), all because of their unity with Christ.

2. Romans 11

In Romans, Paul uses the analogy of an olive tree to explain how Gentile believers are included in God's covenant promises. The olive tree represents God's people, with the root being the patriarchs (Abraham, Isaac, and Jacob) and the natural branches being the Israelites.

> "If some of the natural branches have been broken off, and you, a wild olive shoot, have been grafted in among the others and now share in the nourishing sap from the olive root, do not consider yourself to be superior to those other branches." (Romans 11:17-18)

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He explains that because some of the natural Jewish branches were "hardened" and rejected Christ, Gentile believers have been grafted in. This doesn't mean the Gentiles have replaced Israel, but that they have been brought into the same covenant blessings through faith.

3. Galatians 3:26-29

Paul teaches that faith in Christ is the key to becoming a part of God's people.

> "So in Christ Jesus you are all children of God through faith, for all of you who were baptized into Christ have clothed yourselves with Christ. There is neither Jew nor Gentile, neither slave nor free, nor is there male and female, for you are all one in Christ Jesus. If you belong to Christ, then you are Abraham’s seed, and heirs according to the promise."

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This passage emphasizes that ethnic or social distinctions are no longer a barrier to being included in God's family. By belonging to Christ, both Jews and Gentiles become "Abraham's seed" and inherit the promises of the covenant.