In some church bodies, it is a big deal to be within the Apostolic Succession, for which there are certain rules of succession that apply. Roman Catholicism and Eastern Orthodoxy stand out in this way most prominently, but the Church of the East, the Syrian Orthodox Church, the Coptic Orthodox and Ethiopian Orthodox Churches, and the Armenian Apostolic Church all claim a direct line of episcopal descent from the apostles, as does the Anglican Communion, through the Archbishops of Canterbury, many of whom (as well as some others) were consecrated by the Popes of the Early Middle Ages.
This idea of a succession of church overseers crops up in Irenaeus (d. 202) for the first time, from what I can tell. In Against the Heresies, Irenaeus, Bishop of Lyons in Gaul, calls to the reader’s mind the orthodoxy of the bishops of Rome, who can be traced from Peter (Pope of the Month here) through Linus and Clement (Pope of the Month here) to Eleutherus (c. 174-189).
Irenaeus is writing against various Gnostic sects and uses the apostolic origins of the bishops of Rome to demonstrate the truth of their teaching — if the apostles had secret knowledge, they would have passed it along to their successors. As it is, what is visible from the standard tradition of the Church of Rome in the days from Justin through Hippolytus (about whom read this but also this) is what we think of as ‘orthodoxy’ or, in some scholarly circles, ‘proto-orthodoxy.’ That is — not Gnosticism.
The purpose of the Apostolic Succession for Irenaeus is to demonstrate orthodoxy. The Bishops of Rome believe the following; they trace their teaching and authority from the Apostles; therefore, we can trust them. Tertullian, On the Prescription of Heretics, c. 190, uses the idea in a similar way.
This is important to consider. Today, Anglicans are passed over by the Church of Rome because a nineteenth-century committee decided we are outside of the Apostolic Succession. Anglicans are concerned about the viability of Methodist holy orders because John Wesley stepped outside the Apostolic Succession to promote their movement. The Orthodox at times claim that Protestants in breaking with Rome have removed themselves (ourselves?) from the Apostolic Succession and tradition, explaining the many strange journeys we have taken in the past 500 years.
For Irenaeus and Tertullian, however, Apostolic Succession is not simply a question of the validity of holy orders or whether a gathering of Christians is a true ‘Church’. Their concern with the Apostolic Succession is the guardianship of orthodoxy. We can trust these teachers to be true because we know where their teaching came from — especially important in a semi-oral culture that did most of its teaching orally. The Gnostics claim special knowledge but are distinct from the Apostles’ visible successors.
To take the question of the Methodists, we know where their tradition came from, and — even if their ordinations were irregular and uncanonical — we know that it is, in fact, within the bounds of orthodoxy. This seems to be the point of the earliest attestations of Apostolic Succession. Why, then, do we use it to exclude Methodists from our Communion?