The Catholic Dictionary explains that "Apostolicity" is the mark of the Church which is dependent on her apostolic origin. Since the Church has a threefold purpose in its apostolicity, it must (1) teach the SAME doctrine as that which Christ taught to the Apostles and which the Apostles transmitted to those they contacted in their evangelization mission. This is given to us without change of any sort. Those beliefs which applied at the time in the early church also apply now.
Next (2) the Church's mission and authority is inherited from the Apostles through a legitimate and uninterrupted succession from them to our present bishops and ministers who guarantee this doctrinal purity and apostolic faith. And then (3) that she is the same society as that which the Apostles founded, having the same structure and form. This means that the system of bishops, priests and deacons is also in our hiearchical organization too.
Although many receive their apostolic succession from various Apostles, each region where they journeyed to was different culturally and ethnically. In this diversity we also find variations in liturgy, in local custom, and in the piety of those societies in which it is found. But essentially the faith is the same in each, the government may vary according to whether there is an elected Patriarch, or a Metropolitan Archbishop, a Church Primate, or an Abbas Primus, but it keeps the same faith before it as was with the Apostles who founded those institutions which make up the Church.
We say that the church is "one, holy, catholic and apostolic." By this, from the first characteristic, we mean that the church is "One" even in spite of its many variations. It is "One" Body, that being the Mystical Body of Christ made up of all its christian members. And as the body has various organs and glands, so they work harmoniously together for the good of the entire body. If any malfunction, then the harmony of the body is disrupted. We are told in Scripture that everyone has gifts from the Holy Spirit to be used in the service of the Church, -- some have the gift of prophecy, some the gift of tongues, some the gift of song, etc. but we all use our gifts for the benefit of the entire Body of which we are members and of which we are united into "One." Even as Christ is One with the Father and the Holy Spirit so we are One together.
As members we share the same Sacraments which Our Lord instituted. We must have belief, then we must be baptized, after which we are confirmed in the faith, accepting it with our entire free will. We share in the same consecrated Bread and Wine which we believe are the true "Body and Blood of Christ" in christian mystery. We believe this because Christ Himself said this to the Apostles at the Last Supper. The elements He held in His Hands were changed into the actual Body and Blood He said they were. But the Eucharist is also a Memorial to Christ, an Unbloody Sacrifice which memorializes His Death and Resurrection for us, until He comes again to us on earth. It is this sharing of the Eucharist which keeps us one and we have the benefits of being grafted onto the Vinestock who is Christ Himself.
The other sacraments we have are based upon the first of Baptism which enters us into the church and each succeeding sacrament ministers to our needs as we progress in the Church's Faith and Practice. We have Marriage which unites the husband and wife before God to form a family. We have Holy Orders in which we transmit and consecrate for service members of the Body into the Priesthood, and into the position of "overseer" as Bishops are called. And we have Unction which heals by the anointing with oil and the laying on of hands. We have penance too in which when we repent of our sins and shortcomings, the priest offers forgiveness via God's channel of grace. When that is given after baptism we are assured that our sins are forgiven and we may continue on the path again as before our sins overtook us.
The church also is "holy." In this we mean that it is directed towards holiness, to the improvement of the individual in religious application. And all that it is striving for is for the virtuous, morally righteous, and socially merciful application of the faith to living as God desires us to live both for Him and for other human beings. The Church therefore cannot approve of immorality, degradation, abuse, enslavement, etc. because it is sworn to uphold the principles of our traditional faith. We are meant to love our neighbor, to see that justice is done, and to stem abuses and crime. Because of the "fallen nature" that we inherit by the fall of our first parents in the Garden of Eden, we are assaulted by evil and have an innate weakness towards sin. But we are forbidden to judge others even in weakness. Only God knows the hearts, minds and conditions of each individual, and we must not judge them for their failings of their sins. Instead of judging we offer compassion, acceptance, love and charity. And if someone hates us or is against us, we must pray for them and strive to overcome our own feelings towards them, so that we continue in our striving for perfection in our earthly lives.
The Church is also "catholic" meaning it is "universal" and everywhere present in every country in every city, wherever the faith has been sown, it exists and thrives. Our mission which was the same one given by Our Lord to the Apostles is to preach the Gospel to every nation, to everyone we encounter so that they may know the "good news" that we were set free by the Redemptive Act of Christ the Son of God. And that we are reconciled to God. And how to avail ourselves of this great GIFT that God has given to us in the Sacrifice of His Only Begotten Son. And the last term is "apostolic" which we have explained in the above.
The Apostolic Call was given to all the Apostles equally. All bishops are equal. We give primacy of honor to those who rule the most ancient Sees of the church found at Rome, Constantinople, Alexandria, etc. But each is an independent jurisdiction in that it is self governing. What applies in one jurisdiction, may not apply in others. It is because the bishops must govern them as they see fit, to keep order and to continue to teach what the Apostles always taught. We try to keep each jurisdiction within its own context but sometimes because of the nature of our greatly expanded world view, we have some difficulties which are unique because of the overlapping of territories which in the past did not conflict.
In the history of our faith during the first years we seemed to be united in one fold, but each jurisdiction originates in the efforts of individual Apostles to evangelize various places. And each succession is from a unique source, mainly from the original 12, but even from the second wave which included the 70 who were seen as "like" the Apostles. If one looks at our own Apostolic Succession Tables what is most evident is that there is a Roman line, there is also one through Antioch, one through the Russian Orthodox Church, through the Anglican Church, the Protestant Episcopal Church, the Church of Utrecht, the Old Catholics of Europe and Great Britain, the see of Canterbury and Glastonbury, and the Syrian Church of the East, the Armenians, the Greek Melchites, etc.
Each is a valid line of succession and authorizes the consecrated holder of these lines to keep the same faith as that of the Apostles, to teach the same thing, and to maintain the faith of the people intact. We feel that our bishops apply this most sincerely and effectively for the guiding of the people.