In Romans 6:14b What did Paul mean "..for ye are not under the law, but under grace."?
Fortunately, this is a major theme of the apostle Paul, in all of his writings to Christians, not just to the ones in Rome. By collating all that he said about law and grace, a complete, harmonious grasp of this doctrine of grace is obtained. Such a task, however, is beyond the scope of this site. To do justice to such a collation, a book would be required!
Therefore, I propose to just lift a few key points Paul made to give an outline sketch of what he meant when he wrote that Christians "are not under the law, but under grace." It might be best to start with a clarification of what he meant by this 'law'.
Romans 8:1-2 shows that it is "the law of sin and death". Every human is under that law and must die, as "the wages of sin is death" (6:23). But those who are found "in" Christ by faith have no condemnation because they then "walk not after the flesh, but after the Spirit. For the law of the Spirit of life in Christ Jesus hath made me free from the law of sin and death. For what the law could not do, in that it was weak through the flesh, God sending his own Son in the likeness of sinful flesh, and for sin, condemned sin in the flesh: That the righteousness of the law might be fulfilled in us, who walk not after the flesh, but after the Spirit" (8:1-4).
This means that no amount of law-keeping can avoid God's righteous condemnation, for the law exposes our sinfulness, leaving us without a leg to stand on. But God graciously sent his only-begotten Son to deliver those with faith in him from all condemnation. They are then under the law of the Spirit. And it is faith that is key to walking (living) by the Spirit instead of walking in the flesh (legalism). The whole of Romans chapter 8 details this spiritual 'adoption', so do read it all.
Paul consistently explained this key doctrine of being liberated from legalism (as a means of obtaining God's forgiveness) and being transferred into God's kingdom by the faith of Christ - by grace. Take Galatians chapters 2 & 4. Snippets, to whet your appetite to read all those chapters:
"Knowing that a man is not justified by the works of the law, but by the faith of Jesus Christ... and not by the works of the law; for by the works of the law shall no flesh be justified... for if righteousness come by the law, then Christ is dead in vain... For as many as are of the works of the law are under the curse... The just shall live by faith. And the law is not of faith...
If there had been a law given which could have given life, verily righteousness should have been by the law. But the scripture hath concluded all under sin, that the promise by faith of Jesus Christ might be given to them that believe..." (Galatians 2:16 - 3:29)
Paul goes on to show that Christ came to redeem those under the law "that we might receive the adoption of sons". That is the theme of Romans chapter 8, the indwelling Holy Spirit assuring those with the faith of Christ that they can call God, "Abba: Father". Paul goes on here to add, "But if ye be led of the Spirit, ye are not under the law".
This is supported in 1 Corinthians 15:56: "The sting of death is sin; and the strength of sin is the law, but thanks be to God, which giveth us the victory through our Lord Jesus Christ." That victory is the gift of grace, freely and undeservedly given. Paul reinforces that in Colossians 2:12-23 (do read it all). I just stress these snippets, that Christ is:
"Blotting out the handwriting of ordinances that was against us, which was contrary to us, and took it out of the way, nailing it to the cross."
The contrast Paul keeps making between bondage to the law, with freedom in the Spirit, shows the impossibility of trying to use both. We will fall between two stools if we try that. We have to place all our faith in the finished work of Christ on the cross, and stop trying to perform our own works as some kind of "supplementary" or supportive means of "maintaining" God's free pardon. We either walk in the grace of God by faith (the law of the Spirit), or we don't. That's what Paul meant when he said that Christians "are not under the law, but under grace."
Refer also to 2 Corinthians 3:6 -
The letter killeth, but the Spirit giveth life
Grace perfects the Law in that the Law made sin plain, but did not give the means to overcome it. This was remedied though the Lord's grace.
There was no Spirit present to assist, nor any baptism of power to mortify. (John 7:39) But as some horse (Plato Phædr. §74) that answereth not the rein, it ran indeed, but made frequent slips, the Law meanwhile announcing what was to be done and what not, yet not conveying into those in the race anything over and above exhortation by means of words. But when Christ had come, the effort became afterwards more easy, and therefore we had a more distant goal (μείζονα τὰ σκάμματα) set us, in that the assistance we had given us was greater. Wherefore also Christ saith, Except your righteousness shall exceed the righteousness of the Scribes and Pharisees, ye shall in no case enter into the Kingdom of Heaven (Matthew 5:20). But this he says more clearly in the sequel. But at present he alludes here briefly to it, to show that unless we stoop down very low to it, sin will not get the better of us. For it is not the Law only that exhorteth us, but grace too which also remitted our former sins, and secures us against future ones.
Abiding with the Law of Moses only checked the sin and sinful drives that beset humanity since Adam’s fall, but it failed perfectly to introduce ontological, essential healing and restoring change in human nature.
Through Incarnation, Death and Resurrection of God, the eternal Logos of the Father, came to humans such an intensity of divine presence and intimacy, such powerful working of His operation (forgive me this tautology) which is called “grace” in a less technically-theological language, as to make it possible to thoroughly change and restore human fallenness, giving man an authority of becoming “son of God” (John 1:12), through His natural, only-begotten, co-eternal Son.
Being “subject to grace” means voluntarily to embrace through faith this logic and dynamism of the salvific working of God in us, which means conversion and change of lifestyle in subjection to this logic.