In Matthew 16:18, Jesus promised to build a church. In Acts 2:47, Luke tells us that people were being added to that church. Thus, we can conclude that Jesus built his church sometime between his promise in Matthew 16 and Luke's statement in Acts 2. Indeed, a closer study of the events in Acts 2 reveals that the Lord's church was established on that first day of Pentecost following the Lord's resurrection when Peter preached the first gospel sermon. That church is the church of Christ.
A common misconception about the church of Christ is that "The Church of Christ" is our name. It is not. The "church of Christ" is our description. We are the church that belongs to Christ, that was established by Christ, that was built by Christ, and that was bought by Christ. It is not our church; it is his church. We are not voted into the church by men, and we do not join a church the way some might join a country club. Instead, God adds us to his church when we obey his gospel.
Are those in the church of Christ the only people who are going to be saved? Of course they are! God adds people to his church when they are saved. If you are not in the Lord's church, then you are not saved. If you are saved, then you are in the Lord's church. To be saved outside of the church of Christ is to be saved outside of the body of Christ - and that can never happen. Jesus is not just a way to the Father; he is the way to the Father. As Jesus said in John 14:6 , "I am the way, the truth, and the life: no man cometh unto the Father, but by me."
Thus, the real question is not what is the church of Christ, but is rather how do you become part of the church of Christ? Peter answered that question in Acts 2:38 on the very day that the church was established. And what did Peter tell us to do? Did he tell us to make Jesus Lord of our lives? No. In fact, in Acts 2:36, Peter told us that Jesus is already Lord of everybody's life! We obey Jesus not to make him Lord, but because he is already Lord. To be saved by the Lord and become part of the Lord's church, we must obey the gospel of the Lord. We must, as Peter said in Acts 2:38, "Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost." Anyone who gives you a different answer to the question in Acts 2:37 is not proclaiming the gospel of Jesus Christ.
There is one church of Christ, and that church is not man-made. It is not our church, it is the Lord's church. We do not join it or add people to it or vote people into it. Instead, God adds people to his church when they are saved. That church is not the Catholic church, it did not arise from the Reformation movement, and it did not arise from the Restoration movement. It is the church that Jesus promised to establish in Matthew 16 and that was established in Acts 2. If you are a member of something else or something more or something less, then you are not serving God according to his plan or according to his will. He wants you to be a Christian and only a Christian, wearing only the name of his Son, Jesus Christ, who is the head and the savior of the church, his body.