Apostolic succession is a doctrine accepted by some, but not all Christian denominations. This doctrine states that a person must have specific authority to administer Christian holy ordinances (for example the Eucharist or baptism) and rites, that only a person with this authority can give it (via ordination or another method) to another, and that this line of succession must be traced back to the Apostles, who according to the Bible, held it anciently. (See Matthew 16:18-19, 18:18-20.) Without this divine sanction, ordinances or rites performed by humans are invalid and not recognized by God.
Those denominations that accept this doctrine typically have a way of asserting their claim to the apostolic succession, typically to the exclusion of other denominations from the same claim. Any denomination which accepts the doctrine without laying claim to the authority typically awaits a restoration of the authority by other authorized ministers. This second stance is very uncommon today, but was more common during the time of the Reformation, especially later in the 1700's and 1800's. In addition, many individuals during the Reformation (reformers and theologians) recognized the validity of the doctrine without laying claim to the authority themselves to administer in the things of God. Noteworthy is a comment from Charles Wesley, who broke theologically with his brother John over the authority question:
How easily are bishops made
By man or woman's whim:
Wesley his hands on Coke hath laid,
But who laid hands on him?
In modern times four major groups or denominations acknowledge the doctrine of apostolic succession and lay claim to the priesthood of God:
1. The Holy Roman Apostolic Church (Catholic)
2. The Eastern Orthodox Communion (Eastern Orthodox)
3. The Oriental Orthodox Communion (Oriental Orthodox)
3. The Church of Jesus Christ of Latter-day Saints (Mormon)
The Orthodox and Roman Catholic churches claim that the office of bishop was the greatest in the primitive church, and that bishops were authorized to perform ordinances and ordain new priests and bishops. This authority was passed down from the Apostolic Age to modern times without a break in the line of succession. There are many different opinions and rulings on the validity of ordinations in various splinter churches and groups, and the Roman Catholic Church is generally more ready to recognize foreign ordinations (based on a valid line of the episcopate or bishop's office) than the Orthodox Communions, especially the Oriental. The Roman Catholics claim further that Peter was the first bishop of Rome and that therefore the bishop of Rome holds preeminence over all other bishops and is the Vicar of Christ on Earth, the Lord's personal representative. This view is not shared by the Orthodox churches.
On a similar note, the Anglican Church or Church of England believes that they also possess the line of apostolic succession (through the Archbishop of Canterbury at the time of the schism) but not that it is essential to salvation. At the time of the split from the Roman church the Church of England declared that the Bible was sufficient for salvation and not dependent on the sanction of "popes and popish persons." (Not a direct quote.)
In contrast, the Latter-day Saints believe in a Great Apostasy, a falling away from Christ's teachings during the second, third, and fourth centuries. During this time (or even before with the death of the Apostles) the apostolic authority was lost and the priesthood taken from the Earth by the Lord, who is said to have been displeased with the corruptions of His doctrine by priests and philosophers. This authority was restored to the Prophet Joseph Smith in 1829 by angelic messengers, specifically John the Baptist and the Apostles Peter, James, and John, who held divine authority anciently. This authority of the apostolic office has continued via the Church's Quorum of the Twelve Apostles in unbroken succession to the present day.
Sources
Quoted in C. Beaufort Moss, The Divisions of Christendom: A Retrospect (See http://justus.anglican.org/resources/pc/cbmoss/divisions.html)
Wikipedia, the free encyclopedia. http://en.wikipedia.org/wiki/Apostolic_succession. Wikimedia Foundation. 12 Aug 2005.
Seminar attended by the author: "Braucht die Evangelische Kirche einen Papst?" 27 June 2005. Evangelische Kirche, Hanau, Germany.