Apostolic Succession was established by the apostles when they selected Matthias to replace Judas (Acts 1:23-26). This succession has continued since that day, and presently applies to the New Testament offices of Bishop (Episkopos), Priest (Presbytero), and Deacon (Diakonos) of the Orthodox Church. This threefold arrangement of apostolic offices is found in the writings of Sts. Ignatius and Clement (1st century). St. Clement, who was Bishop of Rome, wrote that Christ instructed his apostles to appoint bishops to succeed them in their apostolic offices throughout the local communities of the Church. Regarding this topic, St. Iranaeus said that apostolic succession is not merely the result of being appointed by way of an apostolic lineage, but also by following the life practices and doctrines of the apostles.
Apostolic succession is important because it ensures the followers of the Lord Jesus Christ that there are certain teachings worthy of our belief which are not mere innovations or interpretations of individual men, but rather teachings which the Lord himself taught the apostles, the apostles taught the fathers, and which have been handed down from the very beginning. Apostolic succession protects the apostolic faith, and without it, believers are subject to the inventions and private interpretations of men.
The consequences of failing to accept apostolic succession is clearly manifested in those who have chosen a kind of Christianity that is not the Orthodox faith. The heterodox follow self-appointed teachers who do not follow the apostolic doctrines and practices visible in the early church, and instead interpret the faith according to their own contemporary wisdom. Of the Roman Catholics and the Anglicans, it may be said that they have a physical lineage that is apostolic, however not a doctrinal lineage. The fact is that they do not hold to the unchanged Orthodox faith of the early church. Of others, it may be said that they are self-appointed without any regard to apostolic lineage, be it physical, doctrinal, or practical.
Consequently, they regularly fail to understand basic Christian truths. For example, notice how those not protected by true apostolic succession fail to understand the unity of the Trinity (as seen in the penal satisfaction theory). They miss the loving and merciful character of God (as seen in the wrathful God wanting to take vengeance on sinners and on the Son of God in the place of sinners). Notice how they fail to see salvation as deification/Christ-likeness (as seen in juridical imputation). Not to mention topics such as the mysteries/sacraments, ecclesiology, eschatology, and many other basic doctrines of apostolic Christianity. Apostolic succession protects the apostolic faith, and without it, believers are subject to the inventions and private interpretations of men.