I had a roommate about 25 years ago who was a Church of Christ member and I learned quite a lot from him about the peculiarities of their beliefs and the different schisms that had arisen among different congregations. The motto he liked to use was, "We speak where the Bible speaks, and are silent where the Bible is silent."
Their founder were Scottish Presbyterian ministers, a father and son team, Thomas and Alexander Campbell who came to the US sometime in the early 1800s and becoming upset at the disunity among all the denominations decided to start a movement back to a more "pure" form of New Testament Christianity. To do that they would throw away all creed books and catechisms etc. and use only the Bible as their guide. I found this link on Google with details of Alexander Campbells life:
http://www.therestorationmovement.com/cmbla.htmUnfortunately while it sounded good in theory, and they started out as one united group, they found like others before them that disagreements arose as to exactly how to interpret the Bible.
There are several different faction within the Church of Christ from what I understand.
Generally they believe that the only practices which are authorized for worship are practices for which they can find authorization for from the New Testament. Authorization can come from a direct command by Jesus or the Apostles or by an approved example set by the practices of the New Testament Christians under the guidance of the divinely inspired apostles.
However, there have been disputes within the Church as to what constitutes an example and how strictly this should be interpreted and this has led to Churches disfellowshipping each other over various aspects of doctrine and worship practices.
For example, since there were no examples of Christians playing music in Church services from the New Testament and there are no commands from Jesus or the Apostles telling Christians to play instruments in worship to God, they should only use a capella (non-instrumental) music in worship services. Playing an instrument for your own amusement at home was not an issue or problem.
Some time ago (it could have been as far back as the early 1900s, not sure when exactly) there was a faction within the Church that decided that playing musical instruments as accompaniment to singing would not piss off Jesus and they went ahead and introduced instrumental music to their services. In the US that group mostly became known as the "Christian Church" to distinguish themselves from the non-instrumental music folks which held on to the name Church of Christ, while in Canada (and in some parts of the US) they maintained the designation "Church of Christ." So in Canada there was the instrumental music Church of Christ and a non-instrumental music Church of Christ. The non-instrumental music folks would not fellowship the instrumental music Church of Christ as they felt the instrumental music folks were acting outside of Biblical authority.
The non-instrumental music CoC or "Christian Church" folks in the US generally had a more "liberal" attitude and also introduced things like Christmas services etc. which the stricter non-instrumental types could not accept either (no examples of Christmas being celebrated in the New Testament Church).
There was another schism that developed around the 1960s in the non-instrumental Church of Christ. There arose a dispute as to whether Church funds could be used for building things like Church gymnasiums for the young folks or dining halls etc for social meetings among Church members. One group said this was not permitted (again,no examples of New Testament Churches having basketball courts are there?) and the other group, although still sticking firmly to the a capella music angle, said that they really didn't think Jesus would mind if they built a dining hall and a Church kitchen next to the auditorium. So again the congregations of the more conservative mindset split off from and refused to fellowship the "liberals" who had Church kitchens, fellowship halls and gymnasiums.
In the city I was in there were 3 different Churches of Christ congregations and neither one talked to the other. There was an instrumental music Church of Christ, a "liberal" Church of Christ and an "anti" Church of Christ. The anti-Church of Christ were the most conservative ones and were given that name "anti" by the liberal Churches of Christ because supposedly they were against (or "anti") everything.
Now within the anti-Churches of Christ there were other somewhat more minor schisms as well. There was the "one cupper" movement and the "no-located preacher" movement which generally went together. The one cuppers believed that since in the Biblical example of the Lord's Supper there was only one cup used for the wine, it would be unscriptural not to use just one cup in the communion service today, i.e. you couldn't use those trays which have the tiny individual cups for each person partaking.
Also following Biblical precedent, the Churches of Christ had preachers or evangelists to preach the sermons and to be responsible for preaching the word to the surrounding neighborhood or town. Now in the Bible the pattern was for Paul, Peter and the Apostles not to live in a fixed location for very long, but to be wondering around the Mediterranean staying awhile in various cities and towns to preach the word and then move on. So some of the anti-CoC churches decided that it would be unscriptural to pay a preacher to set up shop and work with one Church for any extended period of time, rather they should use itinerant preachers that went from town to town to do evangelizing and preaching to the lost and the elders and deacons etc should pitch in more with preaching and teaching the Sunday message if there wasn't an official preacher on hand to do that.
From what I remember the one cuppers and no-located preacher folks would still fellowship the other anti-CoCs even if they didn't go along with their particular obsession to be true to the word of God, and the one cuppers were definitely the smallest of the factions. They had some congregations out in very small towns in the British Columbia interior, if I remember correctly.
As others have mentioned, they believed water baptism was necessary for salvation (just sprinkling water on the forehead wouldn't do, it had to be a complete immersion) and they did not believe in modern day miracles like Benny Hinn, Oral Roberts etc. They believed that miracles were allowed by
God to confirm that Jesus and the Apostles after him were divinely inspired in their work, but once the Apostles had died and left the written scriptures, there was no longer a need for miracles and therefore they ended with the deaths of the Apostles.
It was really quite fascinating to see how, with the best of intentions, people would still come to loggerheads over matters of interpreting scripture, and each faction would be sure their own interpretation was correct while the other factions were in danger of hell fire for falling away from the truth and allowing the devil to lead them astray to act outside of Biblical authority.