Huge question and one that could arguably be answered, "At the beginning of time." John 1:1 in the New Testament says that, "In the beginning was the Word, and the Word was with God, and the Word was God." Verse 2 continues: "He was with God in the beginning."
The name "Christ" comes from the Old Testament. The Hebrew word is "mashiach," (Messiah) which meant "the anointed." "Christos" is derived from the Greek word meaning "to anoint."
The word "Christian" was coined as an insult to the Jewish believers, calling them "Little Christians." The believers were not offended, however. On the contrary, they turned around its intended use of derision and made the name "Christian" a badge of honor, since to be called by that name identified them with Jesus, the Christ. The name stuck, although used very rarely in the New Testament (Acts 11:26; 26:28 and 1 Peter 4:16). Acts 11:26 notes that the new community of believers in Jesus were first called Christians in Antioch. This was during one of three missionary trips Paul made through that city. Trip one was c. A.D. 46-48. Trip two was c. A.D. 50-52. Trip three was A.D. 53-57.
Now to confuse the issue, the belief has little to do with the name. Christians previously called themselves disciples, believers, saints, and bretheren. The Jews called themselves Nazarenes (see Acts 24:5; Mark 14:67), which was again an attempt at an insult to Jesus' hometown of Nazareth. However, in the Old Testament, a Nazarite was one who demonstrated total consecration to the Lord.
Go back to my first paragraph, and you can see that the belief began with God; the group called Christians began in Antioch. Arguably, the "Church" is as old as God's relationship with Man.
The Quest Study Bible, NIV, copyright 1994
Who's Who in the Bible, Paul D. Gardner, Editor, copyright 1995
Both published by Zondervan Publishing House