II Corinthians 3, The New Covenant

As Paul defends his ministry, he moves on to describe in more detail the work of the Holy Spirit in the lives of believers.

2 Corinthians 3: 1-5, You are our recommendation letter!
Are we beginning to commend ourselves again? Or do we need, like some people, letters of recommendation to you or from you? You yourselves are our letter, written on our hearts, known and read by everybody.

You show that you are a letter from Christ, the result of our ministry, written not with ink but with the Spirit of the living God, not on tablets of stone but on tablets of human hearts. Such confidence as this is ours through Christ before God. Not that we are competent in ourselves to claim anything for ourselves, but our competence comes from God.

Paul is aware that he has been defensive.  Here he says, "So you -- your lives -- are my defense!" It has been Christ, working through Paul, that has created this good work in Corinth.

2 Corinthians 3: 6-11, The New Covenant 
He has made us competent as ministers of a new covenant--not of the letter but of the Spirit; for the letter kills, but the Spirit gives life. Now if the ministry that brought death, which was engraved in letters on stone, came with glory, so that the Israelites could not look steadily at the face of Moses because of its glory, fading though it was, will not the ministry of the Spirit be even more glorious? If the ministry that condemns men is glorious, how much more glorious is the ministry that brings righteousness! For what was glorious has no glory now in comparison with the surpassing glory. And if what was fading away came with glory, how much greater is the glory of that which lasts!

The New Covenant writes on our hearts (Jeremiah 31: 31-34) and is more powerful than even the bright Old Testament law given by Moses.

2 Corinthians 3: 12-15, The Veil
Therefore, since we have such a hope, we are very bold. We are not like Moses, who would put a veil over his face to keep the Israelites from gazing at it while the radiance was fading away. But their minds were made dull, for to this day the same veil remains when the old covenant is read. It has not been removed, because only in Christ is it taken away. Even to this day when Moses is read, a veil covers their hearts.

Here Paul pulls up imagery from Moses's encounter with God in Exodus 34 (Exodus 34: 33-35.) The veil prevents direct communication with God and prevents understanding of spiritual things. Some Jewish resistance to the Messiah is the result, says Paul, of a "veil" between them and God.

2 Corinthians 3: 16-18, The Spirit removes the veil
But whenever anyone turns to the Lord, the veil is taken away. Now the Lord is the Spirit, and where the Spirit of the Lord is, there is freedom. And we, who with unveiled faces all reflect the Lord's glory, are being transformed into his likeness with ever-increasing glory, which comes from the Lord, who is the Spirit.

The "veil" separating us from God is removed by the Holy Spirit, which Paul identifies as coming from God and identifies as being God.