What does Paul mean when he says that we are no longer under the law but under grace? Romans 6:14:15

14 For sin shall not have dominion over you: for ye are not under the law, but under grace.
15 What then? shall we sin, because we are not under the law, but under grace? God forbid.

What does Paul mean when he says 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.