The one church can be viewed in different ways, and it is helpful in order to gain a full-orbed understanding of it to consider it in these different ways.
The Invisible Church: This is the church, the Body of Christ, as it is viewed by God. It is called invisible because this view of the church is not accessible to the limited perspective of human beings. God sees the church throughout all time and space, and he sees beyond the pretenses of a false profession of faith, knowing whose faith is (or will be) genuine and whose is not. The Westminster Confession sums it up in this way: "The catholic or universal Church, which is invisible, consists of the whole number of the elect, that have been, are, or shall be gathered into one, under Christ the Head thereof, and is the spouse, the body, the fulness of Him that filleth all in all."
The Visible Church: This is the church as it appears to man. We human beings are bound to time and space, and cannot see into the hearts of men (other than our own). As we look around us, we see various evidences of the presence of Christianity, of the work of the Spirit, of a Christian attitude and profession, of the existence of the Body of Christ, in various places, groups, and individuals. But we can never be sure (outside of ourselves) that those who are making what appears to us to be a credible profession of faith and obedience are really doing so from the heart. There are many who profess Christ but who are not truly sincere in the deepest sense (think of the parable of the sower), some of whom visibly fall away from the faith eventually. The best we can do is examine the professions that people have made, watch for glaring inconsistencies, and employ a judgment of charity to accept as brothers and sisters all those whom such charity allows us to accept. The Westminster Confession describes the visible church in this way: "The visible Church, which is also catholic or universal under the gospel (not confined to one nation as before under the law), consists of all those throughout the world that profess the true religion; and of their children: and is the kingdom of the Lord Jesus Christ, the house and family of God, out of which there is no ordinary possibility of salvation."
Of course, it is the visible church which is the practical form of the church, in that all our dealings in this world with our brothers and sisters in Christ are dependent upon things as they appear to us. It is in the visible church that we find the fellowship of the saints, as well as the "ministry, oracles, and ordinances of God," as the Confession puts it.
Within the concept of the visible church, there are two categories:
The Visible Church De Facto: This refers to an informal evaluation of the presence of the church from the perspective of people, or Christians, in general--that is, it is the church viewed in terms simply of its being in the world as opposed to any question of formal legitimacy or (ecclesiastical) legality. For example, I observe that my neighbor makes a profession of the true religion. I am aware of nothing in his life that clearly contradicts such a profession, and so, in the exercise of a judgment of charity, I accept him as my brother in Christ. A sincere Christian profession, so far as I can see, is consistent with a degree of error and inconsistency. Perhaps my neighbor, for example, is a baptist. I consider him wrong not to baptize his children, but I have no conclusive reason to think he has taken that position out of malice against God and his truth rather than out of a more excusable confusion or ignorance, and so I do not cease to regard him as a brother. On the other hand, if he were living in such a way that the Bible makes clear cannot consist with a regenerated heart, I could not regard his profession as credible.
Similarly, we look around us and we see the communal aspects of the Body of Christ manifested in varying ways and to various degrees. For example, perhaps my neighbor participates in a small house church. There is no formal membership, but there are clearly one or two individuals who function as the teachers of the group, and other spiritual gifts appear to be manifested. I can regard this, so far as I can judge, as a manifestation of the Body of Christ in the world, even if I think it imperfect in various ways. I also look around me and see many other denominations and groups of Christians who, more or less, seem to be manifesting the existence of the church and the doctrines and practices of Christianity.
The Visible Church De Jure: This refer to the church as judged and evaluated from the formal perspective of the legitimate office-bearers and courts of the church in their formal capacity--that is, it is the church viewed according to the rules of formal legality and legitimacy. Christ has given the keys of governance to the legitimate officers and courts of the church. It is their job to constitute the church's formal and legal presence in the world, providing a concrete foundation around which the members of the church are to gather, dispensing the sacraments, teaching the Word of God, etc. It is their job to formally welcome individuals into the church, to dispense church privileges (such as baptism and the Lord's Supper) to members, to exercise spiritual oversight and discipline over them, to formally excommunicate those who need to be cut off from the body, etc.
The legitimate office-bearers and courts of the church have the duty of formally recognizing who the members of the church are, as well as who the officers of the church are. Officers recognize each other by joining into formal church courts--such as church-sessions, presbyteries, national councils, and ecumenical councils. They recognize members by keeping a formal roll of members over whom they are pledged to exercise spiritual authority (and we might distinguish between members who are in partial communion and those who are in full communion with the church). Church members formally recognize these officer-bearers and courts, agreeing to submit to them, and they recognize formally the rest of the body by participating with them in public worship, the sacraments, and the other functions of the church.
On the de jure level, more scrutiny is required of those who would be counted as members and leaders of the church. On the de facto level, anyone who, in the providence of God, is actually exercising a gift of teaching, for example, is exercising a gift of teaching and so is a teacher in the church whom God may choose to use in that capacity. On the formal de jure level, however, there ought to be clearer formal rules regarding whom should be admitted into the role of teacher in the church, and the exercise of such an office needs to be formally recognized by those who have the authority to recognize it. A person cannot simply stand up and declare himself to be a formal teacher in the church without being examined and approved by the leaders of the church. Likewise, on a de facto level, I may regard with a judgment of charity my neighbor who appears, so far as I can tell, to have a credible profession of faith. However, church officers and courts must be more careful in admitting members formally into communion with the church and to the privileges of church membership. In my view, for example, while a baptist may be a true Christian and thus regarded as such by a judgment of charity on the de facto level, a baptist ought not ordinarily to be admitted to formal full communion in the church, on the grounds that his refusal to baptize his children is in violation of the command of God that children are to be treated as insiders in the covenant community. The church cannot judge his motives, whether he be innocent (more or less) in his error; they must require him to conform to the commands of Christ and treat him accordingly if he does not. (There are certainly nuances here that could be better brought out if this was the time and place to do so.) On a de facto level, I may hope and assume that his motives are relatively pure, but the church acting formally must not connive at the outward action of refusing to comply to a command of Christ.
The unity of the church on a de facto level transcends denominational barriers. However, the unity of the church on the formal, de jure level is limited by those barriers. The officers and courts of each denomination have conflicting views of the de jure church. They have a different set of recognized officer-bearers and church courts, and they do not recognize each others' authority. They have a different roll of members. The members of different denominations have a commitment to submit to different officers and courts, and their formal communion with each other is limited by their denominational affiliation. In order to ascertain the correct de jure view of the church, we have to ascertain which are the legitimate officers and courts of the church, but this is precisely where agreement is lacking between denominations. Members and officers of the Orthodox Presbyterian Church, for example, have a different set of officers and courts that are treated as authoritative than do members and officers in the Free Presbyterian Church of Scotland. The views are incompatible, since the denominations are not in formal full communion with each other. Coming to view the church from the proper formal de jure perspective, therefore, requires that we ascertain which denomination actually possesses the authority they all claim. And then we must look at the church from the formal perspective of that denomination. My current position is that the proper denomination from which to view the formal de jure existence of the church is the Free Presbyterian Church of Scotland. See here for more details on the basis of this claim.
More discussion of the issues raised in this article can be found here, as well as in my ongoing series on church authority, church unity, and presbyterian church government (which mostly overlap with each other).