Achaia was the Roman province in which Corinth was located. It was the largest city in southern Greece and the capital of the province. (See: [[rc://en/ta/man/translate/translate-names]])
The Jews took Paul by force to bring Paul before the court. Here "judgment seat" refers to the place where Gallio sat when he made legal decisions in court. AT: "took him so that the governor could judge him at the judgment seat" (See: [[rc://en/ta/man/translate/figs-metonymy]])