This is often a confusing question due to all the cultural and historical baggage that comes with the concept of the church. The New Testament was originally written in Greek which was the language spoken internationally. The English word "church" is the English translation for the original Greek word in the Bible - "ekklesia". "Ekklesia" simply means "assembly". The "church" of Christ, is therefore, the "assembly" belonging to Christ. The "ekklesia" or "church" has nothing to do with a building and everything to do with the people. The Church of Christ, therefore, is the people of Christ which are those who have given up their lives to Jesus as Lord and Christ (Acts 2:36). With or without a building, the churches of Christ have always existed from the time Jesus founded it some 2000 years ago.
Our Non-Denominational Nature
The churches of Christ are not a denomination. There is no home office, no headquarters, and no written creed. The churches of Christ are a fellowship of congregations where each congregation is autonomous. There is no "official list" of congregations in the fellowship. Neither is there a governing board or "membership requirements" for a congregation to be in fellowship other than what is in the Bible. In other words, if a church is practicing and teaching the Bible in all its purity, and not a creed or any such thing, then the Bible demands fellowship, which we extend. The "Church of Christ", then, is not a denominational designation, but a description and identifier of who we are.
Many do not understand that the Lord's church can, should, and must be a church without being a denomination. A denomination is a division or divided part. The only divisions spoken of in scripture are geographical ones which include the churches in Galatia, the churches in Macedonia, the church in Rome, the church in Antioch, etc. Open fellowship existed among these congregations. Preachers such as Apollos, Paul, Timothy, and others, traveled among these churches and worked with them. Hospitality was openly extended to members of all these churches when they traveled. They worked together to preach the Gospel. In the Bible we find simply, "the church-of-Christ". It is necessary to note that this open fellowship was based on doctrinal purity, or pure teaching and practice of the truth. (Galatians 1:6-9; I John 4:1)
Jesus prayed that his followers would be one just as he and the Father were one (John 17:11). Being divided up into denominations with the earthly headquarters, written creeds and earthly founders goes against the wishes of Jesus. Ephesians 4:4ff says there is one body (the church), and one faith. There are not many but one. Our goal is to be as the first century church. Being simply "the church", we have no founder but Jesus, and no headquarters but Heaven. The "Church of Christ" was founded 2000 years ago. That's what we want to be and no more. We step out of denominational ideals, and into simple New Testament Christianity as is prescribed in the Bible.
The name "church of Christ" is not a title (though it is usually used consistently for the benefit of those traveling to identify and differentiate us from a denomination), but rather a description. We use this name because it is found within the scriptures (Romans 16:16), and it gives glory to our founder, Jesus Christ. After all, Jesus did say, "I will build MY church" (Matthew 16:18); and the Bible tells us that Jesus established the church with his own blood (Acts 20:28). Why name His church after a man (Lutheran, Wesley, and Mennonite), or after a form of government (Presbyterian, Episcopal), or even a particular doctrine (Baptist, Methodist), when the Bible simply calls it the churches of Christ, or church of God, or household of God? All of the names used in scripture give glory to God. Instead of focusing on a title, we strive to live up to the "descriptions" of the church in the Bible.
Membership of the Lord's Church
This is another topic that has been confusing to some. Having no creed but the Bible alone, we look to the Bible to determine our terms of membership. As a result, we do not "vote" a person into the congregation. Neither do we have a probationary period, a special class, or any other such thing to become a member. These things are simply not mentioned or even implied in scripture.
Terms of membership into the Lord's church are the same as those for Salvation as seen below. Ephesians 1:1-14 tells us that God has blessed those "In Him" or "In Christ" with every spiritual blessing, with redemption and forgiveness of sins, and with an inheritance of Heaven. So then, when one is saved, he/she becomes part of God's household. This example is clearly seen in Acts 2:37-41. The people heard the message, repented (turned their life over to Christ), were Baptized, and were added to the church (verse 41).
What are the conditions of salvation defined by the Bible?
First we read we are to Hear the word of God, Romans 10:17- "So then faith cometh by hearing, and hearing by the word of God."
Second, we all need to Believe (faith) in Jesus Christ and that his word is true. Hebrews 11:6-" But without faith it is impossible to please him: for he that cometh to god must believe that he is, and that he is a rewarder of them that diligently seek him.
Third, we are to Repent, Acts 2:38- Then Peter said unto them, Repent, and be baptized everyone of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost."
Fourth, we are to Confess, that Jesus Christ is the Son of God Matthew 10:32-33-"whosoever therefore shall confess me before men, him will I confess also before my Father which is in heaven. But whosoever shall deny me before men, him will I also deny before my Father which is in heaven." Another passage is Acts 8:36-37. In this passage we can read that we are to confess Jesus Christ as the Son of God before Men.
Fifth, we are to be Baptized, Romans 6:3-5- "Knew ye not, that so many of us as were baptized into Jesus Christ were, baptized into his death? Therefore we are buried with him by baptism into death: that like as Christ was raised up from the dead by the glory of the Father, even so we also should walk in newness of life. For if we have been planted together in the likeness of his death, we shall be also in the likeness of his resurrection:" In John 3:5 -"Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God." In both these verses we see that we are to be baptized and this is to be done by being buried (immersed) in water.
When one follows the Plan of Salvation as listed above by having faith (belief) in Christ, and that faith leads him/her to repent and turn his/her life over to Christ to make Jesus his/her Lord and Savior, which leads him/her to confess that faith and be baptized in the death burial and resurrection of Jesus for remission of our sins - that person becomes a child of God and a part of the body of Christ, the church. (Acts 2:41-47). The church is simply the "assembly" of those who are saved. We do not and will not impose any man made rules on any Christian to be a member of the Lord's church.