To answer this question in one sentence, they are people who wish to go back to the Bible for their only guide in religious matters, thus restoring in our time the original New Testament church.
If we go back in time to the latter part of the eighteenth century there were different men of differing denominations, each studying the Bible independently of each other, and coming to the same conclusions. These men had a great desire for unity, for this is what Christ prayed for in John 17:20-21: “Neither pray I for these alone, but for them also which shall believe on me through their word; That they all may be one; as thou, Father, art in me, and I in thee, that they also may be one in us: that the world may believe that thou hast sent me.”
They also realized that this unity could never be achieved if we followed after human creeds, synods, councils, opinions, etc. It could only be achieved if we threw off all denominationalism and these human creeds and follow only the Bible.
This does not establish another denomination, but rather it returns to the original. We thus are not a denomination, neither Catholic, Protestant, nor Jewish; but simply members of the church which Christ established, and purchased with His blood.
This also is why we wear the name church of Christ. It is not used as a denominational designation, but as a descriptive term setting forth that the church belongs to (of) Christ. We also recognize that there are other biblical terms, for example, church of God (I Corinthians 1:2); church of the firstborn (Hebrews 12:23); kingdom of heaven (Matthew 16:19); body of Christ (Ephesians4:12); bride of Christ (Ephesians 5). Any term which signifies the church, kingdom, or body belongs to God, Christ. While it cannot be the Lord’s church without the proper name other factors, doctrine, are important.