11 When Cephas came to Antioch, I opposed him to his face, because he stood condemned. 12 For before certain people came from James, he used to eat with the Gentiles. But when they arrived, he began to draw back and separate himself from the Gentiles because he was afraid of those who belonged to the circumcision group. 13 The other Jews joined him in his hypocrisy, so that by their hypocrisy even Barnabas was led astray.
14 When I saw that they were not acting in line with the truth of the gospel, I said to Cephas in front of them all, "You are a Jew, yet you live like a Gentile and not like a Jew. How is it, then, that you force Gentiles to follow Jewish customs? (TNIV)
Here we have the last of Paul's vignettes about his relationship with the church in Jerusalem. Antioch was in Syria and had a very large Jewish population and a significant number of non-Jews who were interested in Judaism and attached themselves in varying degrees to the Jewish synagogues. In all probability, when Christianity came to Antioch, it gained converts of mixed ethnic backgrounds. What's amazing is that the church initially seemed to thrive and did not have Jew/Gentile conflict, especially if they abandoned Jewish food laws. You would expect that it would have been difficult for the (most likely) Jewish majority to give up those distinctives, but it appears that it was not an issue for quite some time.
When Peter arrived at Antioch he went along with the existing practice. He ate at the same table with Gentiles and ate whatever they ate without any resistance. When men from James came, Peter's actions changed. What did the men from James have to say that changed Peter's actions? That's a complicated issue and deserves a post in its own right, so be looking for one soon. At any rate, the central issue here is that Peter acted as a hypocrite in his separation from the Gentiles. One key thing to note is that the Hypocrisei,' the Greek word rendered hypocrite, doesn't mean that Peter 'did something other than what he professed to believe.' It really means something along the lines of 'pretended' (Dunn 125, Fee makes the same point). He acted contrary to what he knew was the truth of the gospel, that Jews and Gentiles were on equal footing and both children of Abraham and heirs of the promise because of the work of Christ on the cross alone. That is why Paul can used such strong language in verse 11. Peter and James knew what the truth of the gospel was, but the message they sent through their actions was that only those who lived like Jews could be part of the people of God. 'For if Gentiles are forced to "live like Jews" on any matter that is based merely on Torah, then the theological result is that their salvation rests ultimately on "doing Torah" not on grace' (Fee 73). Thus Peter stood condemned (and presumably James too).
It's important to notice that Paul never tells us that Peter repented or changed his position. It seems that there was a rift between Paul and Peter (this seems to be smoothed over by Acts 21:17-25?). Thus, Peter's withdrawal at Antioch was probably used by the Teachers in Galatia to support the necessity of the Galatians coming under the Law and to paint Paul in a negative light. Paul told his side of the story so that the Galatians would see that Paul had actually been in the right.
I think that we need to ask then , why did Paul tell us about the meeting in Gal. 2:1-10? What was the point (thanks to danny for raising this question in the comments of my post on the dating of Galatians)? First, he uses it to show that Peter and James were acting against their principles. In theory they agreed with Paul even though they subsequently pursued a different course. I also think that Paul includes it because it sets up this section. The behavior of Peter and James looks much more problematic in light of the previous agreement. This is especially true if we are to identify the meeting in Galatians 2 with the Jerusalem council of Acts 15 because even the issue of dietary restrictions had already been decided.