Hebrews 8

Hebrews 8:1 We have such an high priest, who is set on the right hand of the throne of the Majesty in the heavens

"Romans testified that Christ was 'at the right hand of God' making 'intercession for us' (Rom. 8:34; also v. 27). And Hebrews unfolds the Atonement with the same picture of the Lord 'on the right hand of the throne of the Majesty in the heavens' (Heb. 8:1), living in eternity 'to make intercession for them' (Heb. 7:25). Here is Christ the Advocate, one who walked unscorched through mortal fires. The Advocate is literally the Father's Counselor, who from personal understanding petitions for mercy for mortals. The Petitioner asks not through mere pleading, but because he can boldly certify that he has paid the price of sin. The great truths of modern revelation show why Christ is an effective advocate, for he satisfied justice (Alma 42:14-15) and in trembling pain 'suffered these things for all, that they might not suffer if they would repent' (D&C 19:16)...Thus, Christ literally intervenes between the Father and mankind to produce harmony. He does more than seek peace and understanding-he pays the price necessary to bring forgiveness. He is the contributing Mediator, the effectual Savior." (Understanding Paul [Salt Lake City: Deseret Book Co., 1983], 216.)

Hebrews 8:6 he obtained a more excellent ministry, by how much also he is the mediator of a better covenant

Jeffrey R. Holland
"On those days when we have special need of heaven's help, we would do well to remember one of the titles given to the Savior in the epistle to the Hebrews. Speaking of Jesus' 'more excellent ministry' and why He is 'the mediator of a better covenant' filled with 'better promises,' this author-presumably the Apostle Paul-tells us that through His mediation and Atonement, Christ became 'an high priest of good things to come.'
"Every one of us has times when we need to know things will get better. Moroni spoke of it in the Book of Mormon as 'hope for a better world.' (Ether 12:4) For emotional health and spiritual stamina, everyone needs to be able to look forward to some respite, to something pleasant and renewing and hopeful, whether that blessing be near at hand or still some distance ahead. It is enough just to know we can get there, that however measured or far away, there is the promise of 'good things to come.'
"My declaration is that this is precisely what the gospel of Jesus Christ offers us, especially in times of need. There is help. There is happiness. There really is light at the end of the tunnel. It is the Light of the World, the Bright and Morning Star, the 'light that is endless, that can never be darkened.' It is the very Son of God Himself. In loving praise far beyond Romeo's reach, we say, 'What light through yonder window breaks?' It is the return of hope, and Jesus is the Sun. To any who may be struggling to see that light and find that hope, I say: Hold on. Keep trying. God loves you. Things will improve. Christ comes to you in His 'more excellent ministry' with a future of 'better promises.' He is your 'high priest of good things to come.'" ("An High Priest of Good Things to Come," Ensign, Nov. 1999, 36)

Hebrews 8:8 I will make a new covenant with the house of Israel

Dallin H. Oaks
"This new covenant is frequently mentioned in the scriptures, ancient and modern. Jeremiah prophesied a 'new covenant with the house of Israel' (Jer. 31:31; see also Heb. 8:8). The New Testament teaches that Christ was 'the mediator of a better covenant, which was established upon better promises' (Heb. 8:6). The Prophet Joseph Smith declared that this covenant was not put in force at the time of Christ's mortal ministry because Israel rejected him (see Teachings of the Prophet Joseph Smith, pp. 14-15). In a revelation given the same month the restored Church was organized, the Lord declared, 'I say unto you that all old covenants have I caused to be done away in this thing; and this is a new and an everlasting covenant, even that which was from the beginning' (D&C 22:1).
"The covenant described in these scriptures, made new by its renewal and confirmation in these latter days, refers to our covenant relationship with Jesus Christ. It incorporates the fulness of the gospel (see D&C 66:2; D&C 132:6), which President Joseph Fielding Smith described as 'the sum total of all gospel covenants and obligations' (Doctrines of Salvation, 1:156)." ("Another Testament of Jesus Christ," Ensign, Mar. 1994, 63-64)

Hebrews 8:10 I will put my laws into their mind and write them in their hearts

Have you ever been in the temple and wondered, "why am I hearing the same thing over and over again?" We know that we learn by repetition, but something else is happening. God is fulfilling a promise he made to Jeremiah 2600 years earlier (Jer. 31:33). When you are in the temple, ask yourself if God isn't putting his laws into your mind. Isn't he writing his law in your heart? We can't write down the temple ceremony, but same finger which wrote the law of Moses in tablets of stone can write the law in our minds and hearts in the house of the Lord. Hence is the law, 'written not with ink, but with the Spirit of the living God; not in tables of stone, but in fleshy tables of the heart.' (2 Cor. 3:3, see also Ezek. 11:19-20)

Hebrews 8:11 all shall know me, from the least to the greatest

Orson Pratt
"Now (in the meridian of time) did all Israel and all Judah know the Lord, from the least of them to the greatest of them? Had they no more need to say, every man to his Jewish neighbor, know ye the Lord? Was that the case anciently, when the Lord offered them the covenant of the everlasting Gospel? No; instead of all Israel and all Judah knowing the Lord, from the least to the greatest, they were the very ones that were cut off and lost the privileges of that covenant. But in the latter days when the fullness of the Gentiles is brought in by the proclamation of the Gospel committed by the angel, then is the time that the Lord will renew this covenant, and the same Gospel that he offered to them eighteen hundred years ago, and which they rejected, will be offered to them again, and all Israel will be saved. As it is written-'There shall come out of Zion a deliverer, and shall turn away ungodliness from Jacob.'" (Journal of Discourses, 26 vols. [London: Latter-day Saints' Book Depot, 1854-1886], 18: 178.)
Bruce R. McConkie
"After Israel had been subject to the law for nearly 900 years, the Lord, through Jeremiah, promised to make a new and better covenant with them, a gospel covenant, a covenant which would bring them into his presence so that all might know him. In using this prophecy of Jeremiah to show that Christ brought the new gospel covenant, Paul is doing the same thing in principle as the Mormon Elders do in quoting Biblical passages about the restoration of the gospel in the latter-days.
"...That the glorious and full effect of the covenant promised through Jeremiah will come to pass during the millennium, is shown by the following statement of the Prophet about making one's calling and election sure: 'This principle ought (in its proper place) to be taught, for God hath not revealed anything to Joseph, but what he will make known unto the Twelve, and even the least saint may know all things as fast as he is able to bear them, for the day must come when no man need say to his neighbor, Know ye the Lord; for all shall know him (who remain) from the least to the greatest. How is this to be done? It is to be done by this sealing power, and the other Comforter spoken of, which will be manifest by revelation.' (Teachings, p. 149.)" (Doctrinal New Testament Commentary, 3 vols. [Salt Lake City: Bookcraft, 1965-1973], 3: 180.)

Hebrews 8:12 I will be merciful to their unrighteousness, and their sins...will I remember no more

Boyd K. Packer
"It was through reading the scriptures, and listening, that I could understand, at least in part, the power of the Atonement. Can you imagine how I felt when finally I could see that if I followed whatever conditions the Redeemer had set, I need never endure the agony of being spiritually unclean? Imagine the consoling, liberating, exalting feeling that will come to you when you see the reality of the Atonement and the practical everyday value of it to you individually." ("Washed Clean," Ensign, May 1997, 10)
Boyd K. Packer
"Letters come from those who have made tragic mistakes. They ask, 'Can I ever be forgiven?'
"The answer is yes!
"The gospel teaches us that relief from torment and guilt can be earned through repentance. Save for those few who defect to perdition after having known a fulness, there is no habit, no addiction, no rebellion, no transgression, no offense exempted from the promise of complete forgiveness." ("The Brilliant Morning of Forgiveness," Ensign, Nov. 1995, 19)
Boyd K. Packer
"The discouraging idea that a mistake (or even a series of them) makes it everlastingly too late, does not come from the Lord. He has said that if we will repent, not only will He forgive us our transgressions, but He will forget them and remember our sins no more. (See Isa. 43:25; Heb. 8:12; Heb. 10:17; D&C 58:42; Alma 36:19.) Repentance is like soap; it can wash sin away. Ground-in dirt may take the strong detergent of discipline to get the stains out, but out they will come." ("To Young Women and Men," Ensign, May 1989, 59)