After much debate had taken place, Peter got up and said to them, "My brothers, you are well aware that from early days God made his choice among you that through my mouth the Gentiles would hear the word of the gospel and believe."
Today's Mass reading is verses 7-21 of the above chapter. Verse 7 is quoted above and is uttered during the debate over whether or not Gentiles should be bound by the Mosaic Law. Deciding that Jesus died for all, the Gentiles did not need to be bound by the Law that prepared the way for him.
This also is one of the Scriptual foundations for Papal primacy. Most people associate preaching to the Gentiles with Paul, because he did so much of it, but we see here that it is through Peter's mouth that the Gentiles would hear the Word. So, even when Paul's preaching, this verse tells us he's doing it in union with Peter's mission. (After all, Paul was in the room when Peter said this and didn't disagree, so it's clear Paul understood this to be the case.) So the Christian mission is to be carried out through Peter, or his successor who are now called Popes.