Acts 18:12-17 reads (NRSV):
12 But when Gallio was proconsul of Achaia, the Jews made a united attack on Paul and brought him before the tribunal. 13 They said, “This man is persuading people to worship God in ways that are contrary to the law.” 14 Just as Paul was about to speak, Gallio said to the Jews, “If it were a matter of crime or serious villainy, I would be justified in accepting the complaint of you Jews; 15 but since it is a matter of questions about words and names and your own law, see to it yourselves; I do not wish to be a judge of these matters.” 16 And he dismissed them from the tribunal. 17 Then all of them seized Sosthenes, the official of the synagogue, and beat him in front of the tribunal. But Gallio paid no attention to any of these things.
Paul got in trouble with the Jews of Achaia and they brought him to Gallio, the Roman Proconsul of the province. Gallio determined that whatever Jewish laws Paul may have violated, no civil (Roman) law was broken. So, being a sensible magistrate, he dismissed the case.
Lucius_Junius_Gallio_Annaeanus was quite a distinguished figure. He was the son of Seneca the Elder and the brother of the philosopher/playwright/statesman/… Seneca the Younger, who was later to be advisor to the Emperor Nero (that did not end well). So this episode of Acts is something of a “crossover” with the Roman history of the Julio-Claudian dynasty and with philosophy: The works of Seneca the Younger are an important part of Stoic philosophy, although even in his own time it was thought that he talked the talk more than he walked the walk.
How a discovery about pagan Rome shed light on early Church history reports on contemporary Roman inscriptions which mention Gallio. From them we know that Gallio was Proconsul for one year beginning on May 1, A.D. 51. Hence the events of Acts 18:12-17 took place during that year, giving a us fixed historical marker in Paul’s ministry.