I was asked: “What exactly is the New Covenant, and where is it described in the Bible.”
The short answer is that the New Covenant is mentioned and described in Jeremiah 31:31-34. That passage is quoted in Hebrews 8:8-11.
If you want a passage that gets right to the point in explaining the New Covenant and the difference between it and the Old Covenant, that passage can be found in Acts 2:14-38. If you want a description of the power of the New Covenant, read Acts 2:39 through the end of chapter 4.
Paul explains the difference between the Old and New Covenants in 2 Corinthians 3 as well as in Galatians 4.
The biggest difference between the Old and New Covenants? Perhaps the most important difference is that everyone who enters the New Covenant receives the Spirit of God, as Peter explains in Acts 2:14-38, quoting Joel 2:28-32. Under the Old Covenant, the Holy Spirit was not promised, and only great people like Daniel, David, and others like them received the Holy Spirit.
There are other differences, too. We have the promise of resurrection was never given under the Old Covenant. We have this promise because Jesus rose from the dead. We get complete forgiveness for our past, good and bad, because in baptism God regards us as having died and risen again in Jesus. We really are risen again, or reborn, because he gives us the Spirit and we thus possess the life of God inside of us.
Great promises! Better, they are offered free in Jesus so that all of us can be made together into a holy people, zealous for good works (Titus 2:11-14), and so that we can fulfill the righteous requirement of the Law by living our lives by the Spirit (Rom. 8:1-14).