The Church: A Brief History
The Catholic Church has always claimed to be the one, true Church of Christ. There are many Christian denominations, but the Church has always taught that Christ intended that there be only one church and that it is that church. It is a church to be unified and under the hierarchy of the apostles, which Christ set up.
Unity of the Church in the Bible
St. Paul speaks of the structure of the church and its need for unity. “So then you are...members of the household of God, built upon the foundation of the apostles and prophets, with Christ Jesus Himself as the cornerstone.” - Ephesians 2: 19 – 20. He speaks of the church as a house with a foundation of Jesus and the apostles. There can only be one house, and that house can be identified by its foundation and cornerstone.
Unity of the Church in the Early Church Fathers
The early Christian writers also spoke at length about remaining in the one, true Church and not leaving the fold of Christ.
See that ye all follow the bishop, even as Christ Jesus does the Father, and the presbytery as ye would the apostles. Do ye also reverence the deacons, as those that carry out the appointment of God. Let no man do anything connected with the Church without the bishop. Let that be deemed a proper Eucharist, which is [administered] either by the bishop, or by one to whom he has entrusted it. Wherever the bishop shall appear, there let the multitude also be; even as, wherever Jesus Christ is, there is the Catholic Church. - Ignatius of Antioch Epistle to the Smyrneans 8:2
Whence you ought to know that the bishop is in the Church, and the Church in the bishop; and if any one be not with the bishop, that he is not in the Church, and that those flatter themselves in vain who creep in, not having peace with God's priests, and think that they communicate secretly with some; while the Church, which is Catholic and one, is not cut nor divided, but is indeed connected and bound together by the cement of priests who cohere with one another. - Cyprian To Florentius Epistle 66/67
[T]he Article, In one Holy Catholic Church,' on which, though one might say many things, we will speak but briefly. It is called Catholic then because it extends over all the world, from one end of the earth to the other; and because it teaches universally and completely one and all the doctrines which ought to come to men's knowledge, concerning things both visible and invisible, heavenly and earthly… for this cause the Faith has securely delivered to thee now the Article, And in one Holy Catholic Church;' that thou mayest avoid their wretched meetings, and ever abide with the Holy Church Catholic in which thou wast regenerated. And if ever thou art sojourning in cities, inquire not simply where the Lord's House is (for the other sects of the profane also attempt to call their own dens houses of the Lord), nor merely where the Church is, but where is the Catholic Church. For this is the peculiar name of this Holy Church, the mother of us all, which is the spouse of our Lord Jesus Christ, the Only-begotten Son of God.” - Cyril of Jerusalem Catechetical Lectures, 18:23,26
Let us note that the very tradition, teaching, and faith of the Catholic Church from the beginning, which the Lord gave, was preached by the Apostles, and was preserved by the Fathers. On this was the Church founded; and if anyone departs from this, he neither is nor any longer ought to be called a Christian. - St. Athanasius Four Letters to Serapion of Thmuis
These four authors speak very clearly that it is not enough to simply believe in Christ, but that a person must belong to the Catholic Church and follow those who have been placed in charge, namely the bishops, the successors of the apostles. They also caution against going to other churches which might look like the Catholic Church, but which are not part of the same structure which Jesus founded. It is only by finding and being a part of the one Church of Christ that a person can know he is following God.
In the Bible – Church Founded by Christ/Apostolic
The Catholic Church claims to be the only one founded by Jesus and passed down continuously through the apostles to the present day. It is very clear from the gospels that Jesus founded a church. “And I tell you, you are Peter, and on this rock I will build my church, and the gates of Hades will not prevail against it." - Matthew 16:18. He tells his apostles that he is starting a church with Peter as its rock. It is also very clear from the Acts of the Apostles and the letters of Paul that they were the initial members of a new church and religion. It speaks frequently of the community of believers gathering for worship and mass. The letter of St. Paul to Timothy is a great example of St. Paul recognizing the importance of succession in the line of bishops. Paul himself had ordained Timothy as bishop. "and what you have heard from me through many witnesses entrust to faithful people who will be able to teach others as well." - 2 Timothy 2:2. Paul now told Timothy to pass on what he had received from Paul on to others so that the faith would be preserved.
The early Church Fathers – Founded by Christ/Apostolic
It is particularly evident in the writings of the Early Church Fathers that they understood the importance of the apostolic nature of the Church.
Our Apostles knew through our Lord Jesus Christ that there would be strife for the office of bishop. For this reason, therefore, having received perfect foreknowledge, they appointed those who have already been mentioned, and afterwards added the further provision that, if they should die, other approved men should succeed to their ministry. - Clement of Rome Letter to the Corinthians 44:1
It is possible, then, for everyone in every Church, who may wish to know the truth, to contemplate the tradition of the Apostles which has been made known throughout the whole world. And we are in a position to enumerate those who were instituted bishops by the Apostles, and their successors to our own times: men who neither knew nor taught anything like these heretics rave about. For if the Apostles had known hidden mysteries which they taught to the elite secretly and apart from the rest, they would have handed them down especially to those very ones to whom they were committing the self-same Churches. For surely they wished all those and their successors to be perfect and without reproach, to whom they handed on their authority. - Irenaeus Against Heresies 3:3:1
After the death of the tyrant, the [Apostle John] came back again to Ephesus from the Island of Patmos; and, upon being invited, he went even to the neighboring cities of the pagans, here to appoint bishops, there to set in order whole Churches, and there to ordain to the clerical estate such as were designated by the Spirit. - Clement of Alexandria Who is the Rich Man that is Saved? 42:2
[The apostles] founded churches in every city, from which all the other churches, one after another, derived the tradition of the faith, and the seeds of doctrine, and are every day deriving them, that they may become churches. Indeed, it is on this account only that they will be able to deem themselves apostolic, as being the offspring of apostolic churches. Every sort of thing must necessarily revert to its original for its classification. Therefore the churches, although they are so many and so great, comprise but the one primitive Church, [founded] by the apostles, from which they all [spring]. In this way, all are primitive, and all are apostolic, while they are all proved to be one in unity. - Tertullian Demurrer Against the Heretics 20
Far be it from me to speak adversely of any of these clergy who, in succession from the apostles, confect by their sacred word the Body of Christ and through whose efforts also it is that we are Christians. - Jerome Letters 14:8
These Church Fathers were from the first centuries of the Church. They could easily trace back the lineage of their bishops to the apostles, so they could see very tangibly how their leaders were descended from the apostles. Today, we don't necessarily have that luxury of tracing our bishops all the way back, but we can see that it was a very important part of their understanding of the Church that the bishops be successors of the apostles.