I'm new to this forum, but I've spent my entire life as a member of the Church of Christ, even graduating from Abilene Christian University.
To answer the original question, what is the restoration movement trying to restore? The short answer is "The church, as described in the New Testament".
Do a google search on "The Last Will and Testament of the Springfield Presbytery." It was written by some of the men who started the restoration movement, and will probably answer your question.
And now, for my own version of the history of the RM. (Warning! I am very long-winded!)
Ever since the "Protestant Reformation". it seems like more and more denominations began popping up, calling each other heretics, so on and so forth.
By the time you got to the American frontier in the early 1800s (Which, at the time, was Kentucky), you had so many different "churches" that someone looking for the "one true church" would give themselves a headache trying to figure out what every group that called themselves "Christians" beleived, and trying to sort through it all.
Add to this that the American Frontier was so far removed from traditional centers of culture and learning that classically trained seminarians were scarce, and you get a situation where a lot of people were left with just the bible to figure out how this "Chruch thing" was supposed to work. And what they discovered was that there was a lot of politics, inter-denominational warfare, name calling, and outright hatred among different groups all claiming to follow the same Jesus. So, these independent-minded Americans, with just the bible for their guide, decided to throw out the last 1800 years, and start over. No denominational names, no divisions, no creeds...their idea was "Let's just start with a blank piece of paper and use only what the NT says about church."
However, as with all things done by mortal men, mistakes were made, error creeped in, and sins were commited. Pride crept into the organization. "We're following the NT exactly, the Baptists/Methodists/Pentecostals/Presbyterians/Woeverists aren't. Therefore, they must be wrong." "They're wrong, so their denomination must not be the church Christ established." "They're not going to the "real" church, so they must all be going to hell." And so the progression goes.
There was also disagreement as to what a restored church should look like. Some felt like if the bible did not specifically authorize something, it must not be allowed. Others felt if the bible did not specifically proscribe something, it should be permitted.
Perhaps the most visible example of this is in instrumental worship. Some felt if God had intended for the Chruch to us instruments, he would have specifically authorized it in his scriptures. Others felt if God had considered instrumental music a sin, he would have specifically banned it.
This difference in philosophies led to the split between Christian Churches and Churches of Christ in 1906. (Actually, the split happened long before that. But 1906 ws when everyone recognized it and made it "official"). The idea of only being permitted to practice what was specifically authorized in scripture permiated a lot of Chruches of Christ through most of the 20th century, and led to the rise of all kinds of questionable ideas, such as proscriptions not just on instrumental music, but paid ministers, meals in the church building, Sunday School classes before worship, supporting missionaires, schools, orphanages, ect, that were not a part of your local congregation...the list goes on and on. With no official governing body stating what the CHurches of Christ officially beleive (which I think is a good thing), the degree to which different congragations subscribed to these beleifs varied, and the most restrictive were practiced only by a small but vocal subset of the Churches of Christ, but the Church gradually adopted the mentality that they were the only ones doing it right, and everyone else was going to hell, and in doing so, the Church of Christ essentially turned into the very thing it was founded to erase-just another divisive faction within God's Church at large.
Fortunately, any time you have a large group of people staunchly devoted to studying and following the word of God, the Spirit shows up and teaches people how they should live, and in the last 30 or 40 years, the "us only" mentality within the Church of Christ has begun to fade. I won't lie and say it has completely gone away. but there is a new openness that you wouldn't have found in a Church of Christ in the 1950s. Some Churches of Christ have even begun embracing instrumental music (gasp!). You will now see Churches of Christ using bible study materials writen by "denomination" preachers, which would have been unheard of 50 years ago. I was raised in the Church of Christ, and I'm still a member, and this growing sense of openness and acceptance of other denominations is something I am a firm beleiver in, and a movement I intend to work toward furthering. I see it as the Church of Christ's return to its roots as a group that transcends denominations. In a way, the modern "Non-Denominational" churches you see more and more of are nothing more than congregations participating in their own, private versions of the Restoration Movement, while skipping 150 years of baggage associated with the Stone-Campbell movement.
I was raised by a God-fearing, Bible believing grandmother who did everything in her power to raise me as a Christian. And, God bless her soul, she beleived the Church of Christ was the only Church "doing it right." I remember listening to many of her lectures on what was wrong with denominational Christianity. I was also cursed with an unfortunate ability to think for myself. My grandfather's entire family consisted of very devout Southern Baptists. I remember one visit to a great uncle who lived in Russelville, Arkansas. It was Wednesday night, so we went to church with him. I missed the sign on the door, and participated in an entire bible class assuiming I was in a Church of Christ. After class, we met in the auditorium for worship, and out of the blue clear sky, and organ began playing! I had sat through an entire bible class in a Baptist church, and discovered that they studied the same bible the same way "we" read it.
A few years later, I was sitting in my own familiar Church of Christ classroom, and out youth minister began a series on why the Church of Christ beleived the things it did. I remember well the night he taught about instrumental music. I had, up until that point, always assumed it was a sin because that was what I was brought up beleiving. But after hearing that lesson, I was thinking, "Seriously? That's all you got? We make a big deal about something the bible is silent on?" There I go, thinking for myself again.
My biggest awakening came in college. Despite attending a Church of Christ affiliated university, many of my classmates were from other denominations. I had a friend who was Methodist. I went to church with her once. Now, I could pick apart Methodist beleifs and tell you why I'm not Methodist, but what I remember most about that day was sitting in a room full of people who were desperately longing for God, reading his book, and trying to follow it. And if I think "they" are doing it "wrong," well, it's not my palce to judge, or to say how much grace God will give to cover their mistakes. Especially since I want God's grace for my own mistakes. My room mate was Baptist. One of the guys in my dorm was Catholic. I met people of all different denominations, all trying to find God and follow Jesus.
I came to the conclusion that, once I realize the Church of Christ is not perfect, and needs God's grace, I am compelled to extend that grace to other Christians who are not in the Church of Christ, and accept them as brothers in Christ.
Even saying that, I don't beleive that grace absolves us of the duty of actually studying the bible, and trying to pattern our churches accordingly. However, such study and teaching needs to be done in love, grace and humility, and in such a way that, even if I have deep-seated theological differences with the Methodists that may mean I don't identify with that particular group, I still accept them as brothers and sisters in Christ, and I recognize that whatever sins I see (or think I see) in their church are no worse than my own sins, and no worse than the Church of Christ's sins. Different sins, maybe, but no worse.