Admittedly, I feel completely inadequate to publish an answer. But with all the discussion, SPIN, and rhetoric circulating around Kenneth Starr's acceptance to take over the Presidency of Baylor University and the storyline that he will place membership with the on campus Baptist Church, I felt it was time to cautiously chime in. The answer I will give is mine and not necessarily the answer of every one who holds membership in a congregation claiming to be the Church of Christ
First, allow me to quote from Frank S. Mead, Handbook of Denominations in the United States, 5th Ed, (Nashville:Abingdon Press), 1970. "There is a distinctive plea for unity at the heart of the Churches of Christ -- a unity that is Bible based. It is believed here that the Bible is the "beginning place" in and through which God-fearing people can achieve spiritual oneness" (p 85). "They disclaim being a denomination, but claim to be nondenominational with no headquarters, no governing boards, and no clergy" (p 86). Mead lists numbers of colleges, universities, and lists a few publications in Texas and Tennessee then stresses, "Since all official status in these institutions is lacking, none of them being authorized to speak for the entire church, their conformity in ideas and teachings in all the more remarkable" (p 87). Elsewhere in the article Mead mentions the concept of congregational autonomy with each congregation being governed by her own elders and deacons. (Mead lists, W.E. McClenney, B.W. Stone, and Earl I. West as sources for his information p 238.)
With this article as background let me give MY answer to the title question: "What is the Church of Christ?"
First while consisting of many congregations scattered around the world, the Church of Christ is universally one as she is: 1) the Body of Christ - Eph 1:22-23; 2) the Bride of Christ - Eph 5:21-33; Rev 21:2; 3) the Household of God - 1 Tim 3:15; and 4) the Kingdom of Christ - Col 1:13) among other descriptive terms. Notice that each term is ultimately singular: body, bride, household, and kingdom; thus individual congregations made up of individual Christians are what comprise the universal singular Church of Christ.
I suppose the second point should answer who is a part of this universal Church of Christ? Going back to God's word we find that those who are in Christ by faith have put on Christ and become part of God's family through the promised Seed of Abraham (Gal 3:26-29). Here, I think is a particularly sticky issue. In my past I have made too much of an argument about baptism and not enough about faith. Let me be very precise in my wording; each individual that is a part of the house of God is saved by God's grace through faith (Eph 2:8-10). Without faith man cannot please God (Heb 11:6). However, what is truly faith? Faith is not mere mental acceptance of facts. Faith is trusting obedience. Faith that does not submit to God is not faith. Those that put on Christ by faith in Gal 3:26 were those who were put in Him when by faith submitted to immersion to contact His blood. Paul tells the Romans Christians he was glad they had obeyed from the heart the standard of teaching that saved them (Rom 6:17-18). That standard of teaching that saved them was the same doctrine that Paul taught the Corinthians - (the Gospel - 1 Cor 15:1-5; Rom 1:16). That good news that saved was the sacrifice and resurrection of Jesus as Christ. Paul explains the Roman's faithful obedience to that gospel in Rom 6:3-6. ALL the individuals world-wide who are en christo (in Christ), and ALL the assemblies of those individuals, are the universal Church (Body, Bride, Kingdom, Household) of Christ.
Now as Mead observed, these congregations are autonomous (self-governed). Basically, that means what we do at Parrish may differ in someways from how they do things at Malibu, Baltimore, Nashville, London, Edinburgh, Kiev, or Jasper. We may see some things as acceptable that others do not. Some of the things they accept may not be acceptable here. Sometimes these differences are merely cultural. Sometimes these differences are simple matters of opinion. However, there may be times when we think a Biblical issue is at the center of our differences. When the issue is considered by one or both to be a matter of doctrinal importance, lines of communication should allow for civil discussion. If we come to an impasse, we may choose to limit cooperative fellowship. That should not mean that each think the other is "hell bound and determined." Such should simply mean we choose to work along side of those we feel are more like us.
Ultimately, God and Christ will judge each congregation (consider the Seven churches of Asia in Revelation 2 & 3 - God judged each individually), and they will judge each individual. Maybe that is what Paul had in mind when he wrote, " . . . work our your OWN salvation with fear and trembling" (Phil 2:12, emphasis; SMc). You and I as individuals will stand before God on our own. We will face judgment as to whether we as an individual were in Christ and lived in sanctification and holiness (1 Thess 4:3-7). Each eldership will give account for the congregation they served (Heb 13:17).
I am a Restorationist. I believe that we must all go back to God's word for life and godliness. I believe as many before me that we must set aside denominational nomenclature and return to purely Biblical ideas and principles. I believe there is room for division in opinions or expediency. I believe mutual understanding of Biblical doctrines provides unity whereas disagreements in doctrine limit fellowship. I believe that as long as humanity is involved there will be differences that seem insurmountable. I believe we should teach the truth as we understand truth, allowing for folks to disagree, while continuing to hold fast to healthy spiritual teaching (sound doctrine). I believe in the end of time God will sort out who is and who is not His children. I believe I must do my best to follow God and to teach others what I learn. I cannot force them to agree with me, but I must share what I see is God's plan. To do less would be irresponsible on my part.
Now let the comments flow!