Apostolic succession is, briefly, both the historical series of bishops each anointed by the previous going back to the apostles, and an implied continuity of doctrinal teaching from the time of the apostles to the present.
As held by the Catholic church, this succession also conveys the ability for priests to perform transubstantiation, and discredits other protestant churches who do not follow from this same continuity.
As an apologetic, the "continuity of doctrinal teaching" is usually what is meant. This avoids some common criticism of some protestant denominations and their pure reliance on the bible, as the bible itself is a product of bishops in apostolic succession. The direct lineage back to the apostles also implies an infallibility to the authority of the Catholic church.
Some points for objection or discussion include:
- A long series of followers does not make a teaching true.
- A number of different denominations all hold to have been apostolicly derived, each with different doctrines. (Though believers will likely contend that there of the core doctrines such as the divinity of Christ, transubstantiation, etc, are held by all and therefore true)
- This deflects from the central question of whether or not a god exists. There is still no current evidence to back up this claim.
Basically this amounts to an appeal to popularity, with an added historical connection that feels like it adds more credence to the claim. But in the end the fact remains that all of these bishops in succession could be wrong, intentionally misleading, or some combination of both down through the ages.
- Entry from The Catechism of the Catholic Church