“Then what becomes of our boasting? It is excluded. By what kind of law? By a law of works? No, but by the law of faith.” -Romans 3:27
Paul has already proven in verses 17-24 that because the Jews disobeyed the Law and thus dishonored God causing his name to be blasphemed among the Gentiles, there is nothing for them to boast about regarding their own righteousness.
But here he takes a slightly different approach to drive the final nail in the coffin. There are legitimate things for which the Jews may boast (Romans 9:4-5), but the law of works is not one of them. Boasting is excluded because righteousness is obtained by the law of faith in Christ—apart from the works of the Law.
To say it another way, the Jews may boast in their heritage and special anointing of God to be priests to the world (and all the privileges that come with that) but they do not obtain righteousness any other way than through faith, as the Gentiles do. The only boasting regarding righteousness that is allowed is boasting in the cross of Christ (which he is about to unpack in chapter 4).
Note: Law (noms) means principle rather than Tora.