Paul (Shaul) uses the phrase “under the Law” throughout his letters but not in a uniform or consistent way. According to Galatians, “not under the law” relates to maturity in one’s relationship with God; the Law, which Paul calls a “caretaker” or “tutor,” is not needed for someone who already knows God’s ways (Gal 3:23-25). But in Romans, Paul uses the phrase differently. He states, “For sin shall not be master over you, for you are not under law but under grace” (Rom 6:14).
Based on the above verse, one may reason that the “law” (Greek: νόμος; nomos) is somehow in opposition to “grace” (χάρις; charis). But “grace” means favor or a good disposition towards someone (Hebrew: חֵן; chen), so how can it be opposite to the law (e.g., Rom 3:31; 4:16; 7:7-16; 10:5)? Shaul is not arguing for opposition, but rather highlighting a difference: benefiting from special favor is preferable to being judged by the law.
More, it may seem that “not under law” means “not under Torah,” but the apostle uses the term “law” (νόμος; nomos) in a variety of ways in Romans. For instance, he speaks of what we might call “natural” or “physical” law that governs our bodies and environment — that is, legal systems unrelated to God’s commandments given at Sinai. Furthermore, Paul speaks of “theological” laws that reflect spiritual realities. Consider how Paul refers to both natural and spiritual laws in the same breath: “I see a different law in the members of my body, waging war against the law of my mind and making me a prisoner of the law of sin which is in my members” (Rom 7:23). Here, Paul refers to two different laws — the law of his body vs. the law of his mind — but neither of these “laws” is exactly equivalent to the Law of Moses.
The broader context of Romans 6-8 is about slavery to the “law of sin and death” (τοῦ νόμου τῆς ἁμαρτίας καὶ τοῦ θανάτου), which should not have power in the lives of believers (Rom 8:2). Paul speaks of dying with Messiah as a death to sin, which is followed by a rising to new life (Rom 6:6–7). In this way, Paul calls for freedom from sin, not from the Torah. So when Paul says that we are “not under law but under grace” (Rom 6:14), which law does he have in mind? In this context, it is the law of sin and death, not the Law of Moses.