Earlier in His ministry Jesus said clearly that righteousness suited for entry to heaven needed to exceed Pharisaical righteousness (Matthew 5:20). Jesus spoke of the incompatibility of a patching up old clothes with new cloth to illustrate that which is old needs to be replaced with something new at times (Matthew 9:16). He pointed out that new wine needed new, not old wineskins otherwise the skins would burst. All of this was to say the Old Covenant could not be patched up or the wine of a New Covenant poured into the old wineskins of the Old Covenant. That which is new needs new material. The New Covenant Jesus was ushering in needed New Covenant wineskins (Matthew 9:17). This was not easy for Hebrews to accept.
Paul is making an argument to the Hebrew believers in Jesus for not returning to the rituals and law of the Old Covenant. In the last chapter he spoke of the annulling of the former covenant due to its weakness and unprofitableness to perfect the saint (Heb. 7:18-19). In this last chapter Jesus is presented as the "surety" or down payment, or proof "of a better covenant" (Hebrews 7:22). Now in chapter 8 Paul will elaborate more on why the New Covenant is "better" than the Old Covenant.
Hebrews 8 (NKJV)
8 Now this is the main point of the things we are saying:
We will now be given a summation of what has been argued thus far in Hebrews. It is a summary of why the New Covenant in Christ is better and superior to and makes the Old Covenant obsolete and annulled.
We have such a High Priest, who is seated at the right hand of the throne of the Majesty in the heavens, 2 a Minister of the sanctuary and of the true tabernacle which the Lord erected, and not man.
Jesus is the fulfillment of the Melchizedekian Priesthood promised by God in the Messianic Psalm 110:4. Jesus has provided through the cross a "once for all" (Hebrews 7:27) completely sufficient and superior atoning sacrifice for all sin. Unlike a Levitical priest who can enter the Holy of Holies only once a year and then with fear and trembling, Jesus "is seated at the right hand of the throne of the Majesty in the heavens." Just as an attorney sits when he rests his case before a judge, Jesus is seated and has rested His case for atonement of sin. And He Himself and His wounds are the decisive evidence for His case and through Him our case as well. Jesus is the "Minister of the sanctuary and of the true tabernacle which the Lord erected, and not man." The distance in superiority of the New to the Old Covenant is as vast as the distance from heaven to earth.
One commentator provides a further good summary stating:
The divine oath of Psalm 110:4 [quoted in Hebrews 5:6; 7:17 and 21] which prophesied of the establishment of an eternal Melchizedekian priesthood had now, in the lifetime of those to whom the Epistle to the Hebrews was written, been fulfilled. Such a Priest is not a figment of wishful thinking or speculation. He is now a reality. He is the believer's possession. The waiting is over. Jesus has risen and sits exalted on the right hand of God. He has completed everything necessary for the atonement of the sins of the whole human race and has sat down in God's heavenly sanctuary in the place of preference and blessing. All of God's favor and authority are His to command. He does not stand or bow in the presence of God as a sinful priest must. He is seated, because He is the Son, and His sacrifice is completed. Although He represents men who are upon the earth, He is not ministering in an earthly temple or tabernacle; He deals with spiritual realities in God's heavenly abode.
Because of what Jesus has done on the cross you and I can come and fellowship with the Father even when we have messed up. Even if we've missed or neglected our devotions, even if we've missed church services or some other shortcoming we can come to the Father through faith in the Son Jesus and fellowship in His Presence. Because of what Jesus did on the cross for us, the impediments and obstacles to God's Presence composed of our sins and spiritual shortcomings are obliterated, the veil of any separation caused by sin is torn asunder and we can enter into God's Holy Presence to fellowship in sweet loving peace and security. God truly and graciously loves us into His presence. Because of this we offer sincere and heartfelt praise to God.
3 For every high priest is appointed to offer both gifts and sacrifices. Therefore it is necessary that this One also have something to offer. 4 For if He were on earth, He would not be a priest, since there are priests who offer the gifts according to the law; 5 who serve the copy and shadow of the heavenly things, as Moses was divinely instructed when he was about to make the tabernacle. For He said, “See that you make all things according to the pattern shown you on the mountain.”
The Levitical priests offered "gifts and sacrifices." Jesus offered Himself as the greatest gift and sacrifice. Levitical priests are limited by the law. They "serve the copy and shadow of the heavenly things." In other words the Tabernacle and Temple, Law of sacrifices and Feast Days, these all were only a "shadow" (Greek skia) an interception of light, something behind what the light shines on and lesser than the light source.
This "shadow" is important because it is a "pattern" (Greek typos) or a die, a shape, a statue, a resemblance, a model, imitation, a figure or pattern of something else. "Moses was divinely instructed when he was about to make the tabernacle. For He said, 'See that you make all things according to the pattern shown you on the mountain." It was important that Moses follow Gods' instructions precisely so that the "pattern" or model God gave him would be an accurate presentation of what God was revealing about His plan of sacrifice and salvation from sin.
6 But now He has obtained a more excellent ministry, inasmuch as He is also Mediator of a better covenant, which was established on better promises. 7 For if that first covenant had been faultless, then no place would have been sought for a second.
A team that wins the championship does not fire its coaches. But there are many coaches fired from the teams that didn't win the championship. You don't change a winning combination. You keep changing and trying to improve a combination that doesn't win. Similarly, if the Old Covenant was a "winner" in perfecting the saints it would have been kept, but because it was a loser in this regard a New Covenant was sought to accomplish what the Old Covenant could not accomplish. This "loser" status of the Old Covenant was not any reflection on God, it was a reflection and revelation to humanity of their weakness and shortcomings. The Old Covenant was based on human obedience (e.g. Deut. 4:1). That humanity could not obey the Old Covenant exposes the deficiency in fallen sinful humanity.
8 Because finding fault with them, He says: “Behold, the days are coming, says the Lord, when I will make a new covenant with the house of Israel and with the house of Judah—9 not according to the covenant that I made with their fathers in the day when I took them by the hand to lead them out of the land of Egypt; because they did not continue in My covenant, and I disregarded them, says the Lord.
God found "fault with them." It was fallen, rebellious, sinful humanity that led to the loss under the Old Covenant and the need for a New Covenant. God foresaw this and laid out this as part of His overall plan so that there would be no mistake about the neediness of sinful humanity and the greatness of His grace.
Here we are introduced to the heart of the New Covenant as well as the distinctive difference and superiority of the New Covenant to the Old. The Old Testament prophets Jeremiah (Jeremiah 31) and Ezekiel (Ezekiel 36) both were inspired by God to predict a New Covenant, "not according to the covenant that I made with their fathers in the day when I took them by the hand to lead them out of the land of Egypt; because they did not continue in My covenant, and I disregarded them, says the Lord." God led His people "by the hand," tenderly like a Father holds the hand of their child and leads them. He led them out of the bondage of Egypt but "they did not continue in My covenant." They received the incredible opportunity to walk with God and receive all His blessings but they complained against God and refused to obey Him and so God "disregarded them, says the LORD." They did not enter the Promised Land. They did not enjoy the Presence of God. But God had a plan to fix humanity.
10 For this is the covenant that I will make with the house of Israel after those days, says the Lord: I will put My laws in their mind and write them on their hearts; and I will be their God, and they shall be My people.
Under the Old Covenant the Laws of God were written impressively on stone tablets. But as impressive as these Laws of God were and as impressively as they were delivered (cf. Exodus 19), they were external. But in the New Covenant God would "put My laws in their mind and write them on their hearts; and I will be their God, and they shall be My people." Jon Courson aptly explains, " grace would put into us everything God wants out of us.…" 
The New Covenant is a covenant of grace. It does involve effort from us, but only effort energized by God's grace and the power of the Holy Spirit in a spiritual birthed person (Phil. 2:12). In reality, this New Covenant is "God who works in you both to will and to do for His good pleasure" (Phil. 2:13). As Paul was inspired to write to the Corinthians, "But by the grace of God I am what I am, and His grace toward me was not in vain; but I labored more abundantly than they all, yet not I, but the grace of God which was with me" (1 Cor. 15:10).
The Old Covenant was as effective as making a New Year's resolution; a few weeks after a well intentioned resolution is made, we fail. But the New Covenant is effective because it is based on God's internal empowerment of the spiritually birthed believer in Jesus. The New Covenant is powerful because the Presence of God comes to reside within the believer by the Holy Spirit who indwells them (e.g. Romans 8:9-10; 1 Cor. 3:16; 6:19-20). We are never alone and we are never to attempt to live in our own strength. The Holy Spirit is in us and will empower us to do, by God's grace, what He calls us to do, for His glory, until He returns.
11 None of them shall teach his neighbor, and none his brother, saying, ‘Know the Lord,’ for all shall know Me, from the least of them to the greatest of them.
Here is the outcome of the New Covenant. It involves the presence of God within, the still small voice of the Spirit leading and guiding and assuring the born again believer. The still small voice with which God spoke to Elijah is not available to all who call on God by grace through faith in Jesus (e.g. 1 Kings 19:12). Praise the Lord for His presence and still small voice!
A congregant who hadn't been in church for some time finally returned and went to the pastor to complain that the pastor hadn't sought him out to bring him back to church. The pastor's response was, "That's not my job. That's the job of the Holy Spirit. Under the New Covenant the Holy Spirit reminds you of what God has written on your heart. The believer knows they should be in church and knows they should read their Bible, and knows they should serve the Lord. You knew you should be in church. You should have simply listened and followed the leading of the Spirit."
A problem arises when people insert themselves in the role that is meant for the Holy Spirit. the spouse complains to or injects what they believe are helpful nudges to get their partner to do what they feel God wants them to do. But more often than not such efforts are counterproductive and more aggravating than advancing the cause of Christ. We have to rest people and circumstances in the hands of the Lord. This is part of what we saw in Hebrews 3 - 4. In the New Covenant we can live at rest knowing that the Holy Spirit is always at work. Yes, there will be times when we are called upon to correct or to encourage, but before we do anything we should sensitively seek the leading of the Holy Spirit in what we do.
Jon Courson offers the following insight:
Gang, we don’t need anyone to put rules on us or regulations around us because the Lord lives in us. And that’s the great thing about Christianity. When you’re walking with the Lord, when you’re living in the New Covenant, you don’t have resolutions and regulations, stipulations and obligations. Those are the way of the law.
Instead, you just do what the Lord is writing on your heart, telling you in your mind, whispering in your ear moment by moment.…
“Talk to that guy over there,” or,
“Go to the Mission,” or,
“Make her some chocolate chip cookies.…”
And all you have to do is say, “Okay. Far out!”—and do it. That’s what it means to be a born-again New Covenant, Spirit-led Christian, for whatever God wants from you, by His grace, He will work within you (Philippians 2:13).
Our brothers and sisters in the first-century church were the most radical Christians in all of history. They sold all of their possessions. They spread throughout the world. They lived for the kingdom. But you know what? They didn’t get together and study Hebrews, because Hebrews wasn’t yet written. They didn’t study the theological implications of Romans because Romans wasn’t written. They didn’t scrutinize the teachings of Jesus as recorded in John’s Gospel because John’s Gospel wasn’t written.
They didn’t have the written New Testament—but they did understand the reality of the New Covenant. They obeyed what the Lord was writing in their hearts. And as a result, they turned the world upside down (Acts 17:6). Then, when the New Testament was written and began to circulate through the church, it was a confirmation of what they were already doing because it was the same Lord who had been writing His will for them upon their hearts.
Today, sad to say, many don’t understand the New Covenant. Our Trinity is God the Father, God the Son, and God the Holy Bible. We’ve lost touch with how the Holy Spirit speaks to us moment by moment because we’ve replaced His voice with the written Word. Many churches and organizations study the Bible and are right in their theology— but they’re dead right because theirs is knowledge for knowledge’s sake. The New Testament was never intended to be an esoteric, intellectual, theological trip for people who like to fill notebooks, answer questions, and work on workbooks. That was never the intent of the New Testament writer. What was the intent? To provide a way believers could be confirmed or corrected in what they were already living out as a result of obeying the still, small voice of the Spirit.
The person who’s really used by the Lord is one who simply says, “You’re going to tell me moment by moment what I should do, and, Lord, I will just say, ‘Yes,’ to whatever You say.” A whole lot of people have made the New Testament writings the new law. Like Pharisees searching for jots, tittles, and interesting insights, they fail to see that the Word was simply written to nudge them along in their walk and to confirm the voice of the Lord in their heart.
Please don't use these words to diminish your study of God's word. Please don't abuse these words to rationalize a more lackadaisical and lazy walk with God. These words should do just the opposite. The Spirit inspired scripture and uses scripture to speak to us and confirm His will. The point is that we need to live and walk in the Spirit if we are to realize the purpose and power of God's word. It's not either or but both and. The New Covenant life in the Spirit is Listening to the inner still small voice of the Spirit, confirming it with His inspired word, then walking in obedience to His confirmed word. That is a living way.
The problem is that Christians are students of the word but never graduate to life to apply what they learn. The problem is that our study of scripture doesn't' t lead us into God's presence, it only leads to a class room experience that never produces spiritual fruit. The problem is students of scripture aren't first listening to their Teacher the Holy Spirit. Listen to the Spirit. Confirm His words with His Word.
12 For I will be merciful to their unrighteousness, and their sins and their lawless deeds I will remember no more.”
"For I will be merciful to their unrighteousness." Think about that people. Think about that. "For I will be merciful to their unrighteousness." The New Covenant is God's revelation and fulfillment of what the prophet Micah wrote, "Who is a God like You, pardoning iniquity and passing over transgression of the remnant of His heritage? He does not retain His anger forever, because He delights in mercy. He will again have compassion on us, and will subdue our iniquities. You will casts all our sins into the depths of the sea. You will give truth to Jacob and mercy to Abraham, which You have sworn to our fathers from days of old" (Micah 7:18-20).
13 In that He says, “A new covenant,” He has made the first obsolete. Now what is becoming obsolete and growing old is ready to vanish away.
Twice the Old Covenant is said to be "obsolete" (Greek palaioo) or worn out, decayed, obsolete. It's ready and time for it to "vanish away."It remains useful in that we can look back at it and see the revelation of God laying the ground work for the New Covenant and It's Provider Jesus. But it is no longer to be relied upon. It is old and worn out. It's time to move on to live under the New Covenant.
One commentary states:
Verse 13 gives the writer's concluding remarks on his citation of the new covenant text from Jeremiah 31:31-34. But this only serves as a launching pad for the main discussion of its superiority which is developed in chapters 9 and 10. The main contention at this point was that the announcement of a new covenant and the introduction of the Son necessitated the cancellation of the Mosaic covenant (7:11,18; 8:7). The new and the old are incompatible. They cannot coexist. The new antiquates the old. Being obsolete, the old was near its vanishing point. It was gone—in A.D. 70.
The New Covenant is a beautiful provision of God's grace. It takes into account and overcomes what the Old Covenant did not. You see, under the Old Covenant the spotlight was on our sin. Under the Old Covenant we experienced guilt and failure. Such guilt and failure is a weight that slows us down to a crawl and even sinks and smothers us in the quicksand of sin. But the New Covenant liberates us from sin and guilt. Under the New Covenant we see our sinfulness, our limitations, our shortcomings in all their various forms, but because of God's gracious provision in Christ He is merciful and blesses us, and puts up with us and keeps drawing us closer to Himself. Under the New Covenant God's constant reminder is "their sins and their lawless deeds I will remember no more."
Mercy is not getting what you deserve. Grace is getting what you don't deserve. It is the mercy and grace of God that draws us into His presence.
What makes me want to know God and walk with Him? Why do I start my day wanting to touch base with the Lord and be in the Word? Why am I here. . .? One reason: God’s mercy. I am such a jerk. I have failed so miserably. I have missed the mark so greatly. I have been so inconsistent and stubborn. I’ve been such a sinner. And yet God keeps blessing me; He keeps putting up with me; He keeps allowing me to know Him.
I look at my life, my family, this church, the nation, the world—and I say, “Lord Your mercy is incredible. How good You’ve been to me. I don’t pray the way I could or should. I don’t know as much as I should know at this point in my walk with You. I don’t serve You with the kind of faithfulness You’re worthy of. But, Lord, You just keep blessing. You keep showing mercy. You keep forgiving—and I have no other choice but to want to know You more.”
This is God's provision to come into His presence. Jesus is the One in Whom and through Whom this provision has been delivered.
 Complete Biblical Library Commentary - The Complete Biblical Library – Hebrews-Jude.
 Courson, J. (2003). Jon Courson’s Application Commentary (p. 1481). Nashville, TN: Thomas Nelson.
 Courson, J. (2003). Jon Courson’s Application Commentary (p. 1482). Nashville, TN: Thomas Nelson.
 Complete Biblical Library Commentary - The Complete Biblical Library – Hebrews-Jude.
 Courson, J. (2003). Jon Courson’s Application Commentary (pp. 1482–1483). Nashville, TN: Thomas Nelson.