"On the next Sabbath almost the whole city gathered to hear the word of the Lord. When the Jews saw the crowds, they were filled with jealousy and talked abusively against what Paul was saying." NIV translation
Wouldn't it be awesome to have an entire city turn out to hear about Jesus? When God works in people's hearts this can happen. That's what happened here but the Jewish leaders weren't happy about it. Instead of being happy that people were coming to worship, they saw there influence diminishing and were jealous. The same thing is true today in many cities where churches compete for "members" instead of reaching out to those who do not know Jesus.
"Then Paul and Barnabas answered them boldly: 'We had to speak the word of God to you first. Since you reject it and do not consider yourselves worthy of eternal life, we now turn to the Gentiles. For this is what the Lord has commanded us: "I have made you a light for the Gentiles, that you may bring salvation to the ends of the earth."'" NIV translation
Now, God reveals his purpose for Paul and Barnabas and they turn their attention to the Gentiles. (That is most of us in the world.) Their new goal is to "bring salvation to the ends of the earth". Like Paul and Barnabas, we all (as Christians) are called to share the message of salvation with the world.
"When the Gentiles heard this, they were glad and honored the word of the Lord; and all who were appointed for eternal life believed." NIV translation
The non-Jews gladly accepted the news of Jesus Christ and many believed on him for salvation. The last part of this verse has caused some confusion among Christians. Many take the "all who were appointed" and use it as an excuse for not going and sharing the gospel. The reasoning is that, since God has appointed them, then we do not need to do anything. The thing to understand is that God knows and has appointed those that will be saved but we do not have the wisdom of God and so we do not know all that will be saved. We cannot look at someone and know whether they will come to Christ or not therefore, we must share the gospel with all people and trust God to work.
"But the Jews incited the God-fearing women of high standing and the leading men of the city. They stirred up persecution against Paul and Barnabas, and expelled them from their region." NIV translation
The rich and powerful in the city worked together to get rid of Paul and Barnabas.