What does Paul mean when he says that we are no longer under the law but under grace?

The traditional understanding is that, whereas in under the Old Covenant actions were sinful because they contravened the law, under the New Covenant they are sinful because they contravene grace and love as shown to us in Jesus Christ.

This is explained, for instance, in the Catechism of the Catholic Church:

The New Law is called a law of love because it makes us act out of the love infused by the Holy Spirit, rather than from fear; a law of grace, because it confers the strength of grace to act, by means of faith and the sacraments; a law of freedom, because it sets us free from the ritual and juridical observances of the Old Law, inclines us to act spontaneously by the prompting of charity and, finally, lets us pass from the condition of a servant who “does not know what his master is doing” to that of a friend of Christ — “For all that I have heard from my Father I have made known to you”—or even to the status of son and heir. (CCC 1972)

Under the law, actions were regulated. Under grace, transformation is required of the whole being of the person. We are commanded not to sin not because it contravenes a set of laws but because it contravenes the grace given to us in Jesus Christ.

What does Paul mean when he says that we are no longer under the law but under grace? To better understand the difference we need to look at the outcome of the purpose. The outcome of the purpose of the law is that others are not offended. The outcome of the purpose of grace is that you yourself become not offended.

For sin shall not have dominion over you, for you are not under law but under grace. Romans 6:14 NKJV)

Now the point of this saying is that "sin shall not have dominion over you" meaning that the result of sin shall no longer have control over you. Therefore when one sins against you, the reaction is not based on the desire that sin wants. Because you are follow the laws of grace we are given the power over the reaction to that sin. For the true cause of offense is the truth of the weakness, and acceptance with humility and forgiveness protects the attitude of the self, from becoming worrisome or upset.

What then? Shall we sin because we are not under law but under grace? Certainly not! (Romans 6:15 NKJV)

So to iterate the point of this saying we substitute the results of the law classifications. "What then? Shall we sin because we are no longer under the law that prevents offense but under the law of the permanent attitude that does not get offended.

Paul clearly sees that verse 14 might be misinterpreted to mean that we are no longer under the law therefore we can sin, so he makes it clear in verse 15 that we must not sin even under grace. However this leaves us with the question of what does Paul mean when he says we are no longer under the law but under grace? In the end when everyone becomes mature in the faith, and offense no longer occurs, then the world will have true peace. Then it is promised that GOD WILL COME BACK TO REIGN OVER OUR LIVES.

Then comes the end, when He delivers the kingdom to God the Father, when He puts an end to all rule and all authority and power. (1 Corinthians 15:24 NKJV)

Notice that the Law does not end.

Do not think that I came to destroy the Law or the Prophets. I did not come to destroy but to fulfill. (Matthew 5:17 NKJV)

Because the fulfillment of the law is the removal of pride. Now is the time to get the non-believers who are ruled by their self pride, and are still capable of angry outbursts and the natural desire to fulfill the requirements of the law to turn to Yeshoush. To obtain the Holy Attitude, We walk by faith and not by the denial of reality which leads to anger and bargaining and depression, and if in that state we believe by faith in Jesus that forgiveness leads to acceptance. Then we ourselves become truly spiritual.

Ref: The Stages of Grief