2 Corinthians 3

==2 Corinthians 3:1-3==
1 Do we begin again to commend ourselves? or need we, as some others, epistles of commendation to you, or letters of commendation from you? 2 Ye are our epistle written in our hearts, known and read of all men: 3 Forasmuch as ye are manifestly declared to be the epistle of Christ ministered by us, written not with ink, but with the Spirit of the living God; not in tables of stone, but in fleshy tables of the heart.'
Paul asks the rhetorical question about needing to commend himself to them as if they had just met. The answer is that he and the congregation at Corinth are well known to each other. Because of their work, their love, they are the commendation of Paul to the world. 
The church at Corinth preached the gospel that had been engrafted into their hearts (James 1:21) just as I pointed out in 2 Corinthians 1:21-22. They were not just preaching the Word as an external thing, but were living it and it showed in their community and abroad. 
Paul then contrasts the gospel -- the letter written with the spirit of God on their hearts -- with the law of Moses -- the law written on tables of stone. 
Remember, the opening context in chatper 1:4-7, the trouble they were having with the persecution of the Jews. Here is more of that context, which is found throughout the New Testament. 
Note that spirit should not be capitalized in this verse. This is a spiritual mindset (Romans 7-8) not the third person of the Godhead. It is written with, not by, the spirit of God. See verse 17 for more.
==2 Corinthians 3:4-6==
4 And such trust have we through Christ to God-ward: 5 Not that we are sufficient of ourselves to think any thing as of ourselves; but our sufficiency is of God; 6 Who also hath made us able ministers of the new testament; not of the letter, but of the spirit: for the letter killeth, but the spirit giveth life. 
Paul then talks about the trust he and the church at Corinth have in God rather than in themselves.
They are made ministers of the new testament, the law of Christ. He then continues the comparison between the law of Moses and the law of Christ with the comparison between the letter and the spirit. The letter is metaphorical for the physical nature of the law of Moses. That law was filled with physical rules to follow, physical statutes and ordinances for a physical nation. These physical things were temporary and pointed toward the better, spiritual things that would come after. As such, it could only condemn, not justify. 
Christ's law is of a spiritual nature. It is far superior because it not only is based in spiritual things, but has the capacity to justify men before God. 
They then are ministers of this new, spiritual, better law and need not fear the waning things of the old, physical, inferior law.
==2 Corinthians 3:7-10==
7 But if the ministration of death, written and engraven in stones, was glorious, so that the children of Israel could not stedfastly behold the face of Moses for the glory of his countenance; which glory was to be done away: 8 How shall not the ministration of the spirit be rather glorious? 9 For if the ministration of condemnation be glory, much more doth the ministration of righteousness exceed in glory. 10 For even that which was made glorious had no glory in this respect, by reason of the glory that excelleth.
The ministration of death is a metaphor for the law of Moses (Romans 8:2). This law, this lesser, physical, condemning law written on temporary stone that could break was so glorious that the children of Israel at Sinai could not behold the face of Moses, forcing him to wear a veil. How much more glorious is the greater, spiritual, life-giving law written on the immortal soul? 
==2 Corinthians 3:11==
11 For if that which is done away was glorious, much more that which remaineth is glorious.
Translation - the New Testament is better than the Old. Why then would anyone want to live by the old?
"That which is done away..." The Old Testament is done away with. It ended its authority over individuals at the cross (Col. 2:14; Eph. 2:15; Hebrews). 
The system it created, the Jewish system that it created lingered on as a physical entity, but no one could sin and be condemned by it any more. No one after the cross was required to keep the Sabbath day. No one was required to maintain the dietary laws. No one was required to sacrifice animals. These all ended when Christ fulfilled them as the antitype of OT things on the cross. The Jews just didn't know it yet. 
==2 Corinthians 3:12-15==
12 Seeing then that we have such hope, we use great plainness of speech: 13 And not as Moses, which put a veil over his face, that the children of Israel could not stedfastly look to the end of that which is abolished: 14 But their minds were blinded: for until this day remaineth the same vail untaken away in the reading of the old testament; which vail is done away in Christ. 15 But even unto this day, when Moses is read, the vail is upon their heart.
The things of the OT were barely understood, even by the prophets when seeing the visions. They had to be explained. This is described as a veil between the people and their understanding of the Old Testament. One of the most important things this veil did was prevented the Israelites from seeing that there would be an end to the law of Moses and subsequently an end to their system.
Verses 14 and 15 drive home the point between the ending of the law itself and the ending of the system, the organization that was built on that law. Paul explicitly says that the veil was done away in Christ. The mystery of the Old Testament, the separation of the Jewish people from the rest of the world ended in Christ. He fulfilled the Mosaical types in becoming the antitype. This is already a past event when Paul writes this letter.
However, to that day, even upon the writing of the letter, the Jews are still blinded. They still have that sense of being better, set apart from the rest of the world. They put faith in being physical descendents of Abraham rather than putting their faith in the God of Abraham. It was the last days, the abolishment of their system approached after the fulfillment of their law had come, and they did not want to see it.
==2 Corinthians 3:16==
16 Nevertheless when it shall turn to the Lord, the vail shall be taken away.
When their minds turn to Christ in faithful obedience, the veil is removed and understanding achieved. They come to realize the purpose of the Israelite people and the Mosaical system, realize that it was fulfilled in Christ, and realize that the old law was fulfilled and the old system was about to be destroyed.
==2 Corinthians 3:17==
17 Now the Lord is that Spirit: and where the Spirit of the Lord is, there is liberty.
This is a powerful verse to help us understand verses 1-3 and even chapter 1:21-22. Christ is that spirit of holiness, that mindset. Paul in his last letter to the Corinthians calls this "having the mind of Christ" (1 Cor. 2:16). When this spirit, or mindset is in the Christians of Corinth, when they conformed themselves to Christ by obedience to His will, they would find freedom from sin, freedom from worry about temporary things, freedom from fear of persecution at the hands of the Jews (or anyone else). 
==2 Corinthians 3:18==
18 But we all, with open face beholding as in a glass the glory of the Lord, are changed into the same image from glory to glory, even as by the Spirit of the Lord.
Referring back to 1 Cor. 13, Paul says that the Christians at Corinth behold in the mirror not their own face, but a reflection of Christ, for Christ is in them shining outward brighter than Moses just come from Sinai. It is their actions based on having the mind of Christ that shine forth into the world for the glory of Christ.
In Truth and Love.
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2 Corinthians 12
2 Corinthians 13