Felix was the governor of Cilicia. When Paul got into a knock-down drag-out with the Jerusalem Jews, Felix was the one that the Roman brass took him to in hopes of getting the matter settled once and for all. Paul's Roman passport entitled him to a Roman hearing, and Felix gave it to him. He seems to have listened sympathetically enough and to have had a fairly good understanding of both sides of the issue, since, on the one hand, he already knew about the Christian movement and, on the other, he had a Jewish wife. Under the pretext of awaiting further evidence, he then placed Paul in custody, but went out of the way to see to it that he was well taken care of. He could do what he wanted within reason, and his friends were allowed to supplement his rations from a kosher delicatessen.
The trouble came during a second interview a couple of days later. Felix had summoned him to find out how much his release was worth to him in hard cash, but with his usual tact Paul insisted on discussing justice, self-control, and future judgment instead. "Don't call me. I'll call you," Felix said and sent him back to the pokey. He dropped in on him there from time to time to pursue his original line of inquiry, but Paul never seemed to zero in on what he was after.
With three squares a day, a roof over his head, and plenty of time to write letters, Paul had no major complaints apparently, and as long as Felix didn't spring him, the Jews had no major complaints either. As for Felix himself, after two years he retired on a handsome government pension, leaving the problem of what to do with Paul for his successor to worry about. Felix, of course, means "the happy one" in Latin, and if happiness consists of having your cake and eating it too, he was well named.