"So, what is the 'New Covenant' anyway?"

Great Question! "The New Covenant" or "the covenant of grace" dispenses blessing based upon JESUS CHRIST'S PERFECT PERFORMANCE  UNDER THE LAW during His life on earth (Gal 4:4-5).

His obedience was complete and He was the only man to deserve ALL OF THE BLESSINGS of Deut 28. Through His perfect sacrifice He forever made perfect and pure (took away the sins of) the recipients of this better covenant Heb 10:10-13. When those presently under the first covenant of law abandon it and their own self righteousness (condemned under the law) and put their faith in CHRISTS PERFECT PERFORMANCE and SACRIFICE...His perfect score card is transferred to them FOREVER (the righteousness from God Rom 1:16-18), thus qualifying them for the FULL BLESSING at all times based on CHRIST'S OBEDIENCE. His perfect sacrifice was offered ONE TIME, because it had the ability to permanently purify those who receive it Heb 10:1-4, Heb 10:10-12. Because the recipients of this new covenant have had their sins removed once (past tense) for all time there is no more need for a sacrifice for sins. Because this sacrifice (JESUS!) did it's job the first time, there is no more reminder of the peoples sin under the new covenant Heb 10:2-3. This is a better covenant (ministry of life) with better promises (life and blessing) based on a better foundation (Christ's performance, not man's) Heb 8:6.

THE AMAZING GOSPEL-or "the ministry of reconciliation" 2 Cor 5:18 which is that, IN CHRIST, God is no longer holding the worlds sin against it 2 Cor 5:19. That there is now permanent peace (unhindered un-breakable fellowship, favor and blessing with God) available to all who put their faith in Jesus Christ and receive the ministry of the new covenant. There is blessing, life, peace, and freedom available completely based upon Christ's perfect performance. For all who reject this gift there is condemnation, wrath and eternal punishment (Romans 1:16-18)