As you may know, there are other churches in Australia that call themselves "Church of Christ". Generally, these churches actually belong to the Associated Churches of Christ, which is an association (as the name indicates) and therefore a denomination. We use the name: Church of Christ, as it is a designation found in the Bible (Romans 16:16). However, we like to differentiate ourselves from the Associated Churches of Christ and all other denominations by pointing out that we are "non-denominational".
What is a denomination, and why have we chosen to avoid being one? First, allow us to explain the word denomination in three ways based upon its usage. The first involves an illustration from mathematics, so if you're not a fan of that you may want to skip to the next paragraph. When a number is divided by another number, it becomes a fraction. The number that is being divided (the one that comes on top) is called the numerator, and the number that it is being divided by (the one on bottom) is called a denominator. The word denomination and the word denominator are related in that both imply division has taken place.
The second explanation involves money. In Australia, our currency is called the Australian Dollar. However, when our currency takes paper or coin form, it does so in different denominations, or, in other words, different values. The word denomination therefore implies that there has been a change of values.
The third explanation involves the word itself. One of the word's definitions is, "Identifying word or words by which someone or something is called and classified or distinguished from others" (Wordweb Dictionary). Here we see that the word denomination denotes that a group has chosen a name to signify that it is different from other groups.
When Jesus Christ was still upon this earth, He promised, "I will build my church". You don't have to have a degree in English to see that church is a singular word - Christ said He would only build one church. In fact, when you look into the pages of the New Testament, you see just one church. Yes, there were many congregations, such as the church at Corinth or the church at Galatia, but each of these churches believed, taught, and practiced the same faith (1 Corinthians 4:17).
As you examine the world around you, you see that man has built many churches. Driving down a major road in Melbourne, you might find a Uniting Church, an Anglican Church, a Catholic Church, a Mormon Temple, a Jehovah's Witness Kingdom Hall, and maybe more. While many characteristics of these churches are the same, there are many practices and beliefs that divide them. This is because each of these churches was built when men practiced denomination - giving their individual church a name (sometimes a name from the Bible, sometimes not) and a set of values other than those values found solely in the Bible, thereby creating division. Their values are often expressed in creeds, statements/confessions of faith, manuals, articles of association, and other documents that go beyond what the Bible says.
Jesus did not want His church to be divided. In fact, he prayed that His followers would be as united as He was with God the Father (John 17:20-23). Paul begged the church at Corinth on behalf of the Lord to be completely joined both in thought and in practice (1 Corinthians 1:10). Peter told us that if we were going to wear a name and bring glorabout us 3y to God it needed to be the name Christian (1 Peter 4:16), not a man-made denominational name. Jesus did not want his followers to denominate themselves; He wanted His followers to be united upon the foundation of His authority found in the Bible (Colossians 3:17). In fact, when man picks up man-made tradition, he casts aside God's commandments (Mark 7:6-8).
We call ourselves the church of Christ not because we are "Church of Christ Christians", but because we desire to be the church that belongs to Jesus Christ. Individually, we call ourselves "Christians" because we seek to be identified only as the disciples of Jesus Christ (Acts 11:26). We do not have a creed, statement of faith, manual, or any such document stating our beliefs, because our beliefs are found in the Bible alone. We do our best to speak where the Bible speaks and be silent where the Bible is silent. Further, we seek to do Bible things in Bible ways, and call Bible things by Bible names. This is what "non-denominational" Christianity is all about - bringing the church that Jesus built in the first century into the 21st century.