"Now those who were scattered after the persecution that arose over Stephen traveled as far as Phoenicia, Cyprus, and Antioch, preaching the word to no one but the Jews only. But some of them were men from Cyprus and Cyrene, who, when they had come to Antioch, spoke to the Hellenists, preaching the Lord Jesus. And the hand of the Lord was with them, and a great number believed and turned to the Lord. Then news of these things came to the ears of the church in Jerusalem, and they sent out Barnabas to go as far as Antioch. When he came and had seen the grace of God, he was glad, and encouraged them all that with purpose of heart they should continue with the Lord. For he was a good man, full of the Holy Spirit and of faith. And a great many people were added to the Lord. Then Barnabas departed for Tarsus to seek Saul. And when he had found him, he brought him back to Antioch. So it was that for a whole year they assembled with the church and taught a great many people. And the disciples were first called Christians in Antioch" (Acts 11:19-26).
We learned from Acts 8:4 that the disciples went everywhere preaching the word. At this point in the history of the early church, Peter's preaching to a group of Gentiles was the exception, not the rule. The disciples are focusing their evangelistic efforts upon the Jews only, but that would change soon as Cornelius' conversion became well known. The Lord blessed these disciples and they rapidly grew since a "great number believed and turned to the Lord" among those whom they taught (note that believing is not the same thing as turning to the Lord; the two are distinguishable). Once this good news traveled back to Jerusalem, the great encourager, Barnabas, was sent to strengthen these new followers of Jesus Christ. He traveled to them and was thrilled to see their faith in action. Even today it is exciting to see people give their lives to the Lord and serve Him with all their might! Barnabas reminded these disciples of a central truth of following Christ; namely, that one must continue with the Lord (cf. Matt. 10:22; Rev. 2:10). Such is an easy task when things are going smoothly, but it is not so easy when problems rise up and threaten us. The fact that continuing with the Lord is commanded implies the possibility of failing to do such, which further implies the possibility of apostasy.
Why was Barnabas sent? Likely because of his ability to work with people and build them up in the faith. He was a lot like Jesus, for he went about doing good (cf. Acts 10:38). He was a great man and full of the Holy Spirit and faith. He helped a large number of people obey the gospel, thus being added to the Lord and His church. In other words, as we've seen elsewhere in the book of Acts, these people believed, repented of their sins, confessed Christ, and were immersed in water for the remission of their sins. This pattern still fits today. At the time of faith, one has changed his conviction; at the point of repentance, he has altered his mode of living; when he is baptized, he has changed his relationship to an "in Christ" association (cf. Rom. 6:3,4; Gal. 3:26,27).
Barnabas thought it wise to seek Saul of Tarsus' assistance, so he tracked him down and brought him back to Tarsus to co-labor with him. They taught and encouraged many souls in Antioch for an entire year (this is one example of a "located ministry"). Barnabas and Saul loved to gather together with other disciples. They enjoyed worshiping and working together in the Lord. The descriptive term "Christian" (i.e., follower of Christ) was first given to disciples in Antioch. There is evidence that this name was divinely given (cf. Isa. 62:2). This descriptive term is simple and does not need to be augmented in any way. To affirm that one is a Christian should be sufficient; more specific terminology should not be needed for those who believe and follow God's word. One never reads about Lutherans, Presbyterians, Methodists, etc. in the New Testament--only Christians!