When Cephas came to Antioch: Paul confronting Peter in the city of Antioch is not in the book of Acts, but it likely happened after Paul and Barnabas came back from their first journey to tell people about Jesus (Acts 14:26–28). Paul probably wrote this letter soon after. (The apostle Paul often used the name Cephas when writing about Peter in his letters. Cephas is an Aramaic word that means "rock" or "stone." In Greek, this same name is Petros, which we know in English as "Peter.")
he stood condemned: Peter's actions did not match what he knew—that God accepts non-Jews through faith, not by following the law (see Acts 10–11).
Paul had to confront Peter directly. Paul wanted to protect the good news from being changed (Galatians 2:21). This meant publicly showing that Peter's public action was wrong (compare 1 Timothy 5:20).