Acts 11:15-17 (NIV)
"As I began to speak, the Holy Spirit came on them as he had come on us at the beginning. Then I remembered what the Lord had said: 'John baptized with water, but you will be baptized with the Holy Spirit.' So if God gave them the same gift as he gave us, who believed in the Lord Jesus Christ, who was I to think that I could oppose God?"
Peter and the other apostles were all Jews. Jesus was a Jew. The Jews were God's chosen people. The tradition and history of the Jews was, for the Jew, their life--their reason for existing. From our perspective, the good news of Jesus Christ means anyone can be saved by coming to Christ. For the Jew, it was not this simple. They thought you had to be a Jew to receive salvation. They were very thick-headed about this. The Lord had to get Peter's attention with the vision of the unclean animals coming down in a sheet from heaven saying "Arise, kill and eat."
The final acceptance of the message that the gentiles were included in God's plan did not come until God demonstrated it with a sign. That sign was the miraculous gift of speaking in tongues by the Holy Spirit. That did the trick because this was the same sign Peter and the others had been given when the Holy Spirit had first been sent on the day of Pentecost. This was a huge hurdle for the Jews to overcome. It took some real attention getting signs from God. This is the purpose of the gift of speaking in tongues. We don't even consider it an issue today which is why this particular gift is no longer necessary.