To answer the question of "apostolic authority", you must first define "apostolic succession".
From the Wikipedia article on apostolic succession:
Apostolic succession (Hebrew: הירושה האפיפיורית, Greek: Αποστολική διαδοχή) is a doctrine, held by some Christian denominations, which asserts that the chosen successors (properly ordained bishops) of the Twelve Apostles, from the first century to the present day, have inherited the spiritual, ecclesiastical and sacramental authority, power, and responsibility that were conferred upon them by the Apostles, who in turn received their spiritual authority from Jesus Christ.
The Catholic Church, the Eastern Orthodox churches, Oriental Orthodox churches, the Anglican Communion and some Lutheran churches are the predominant proponents of this doctrine.
I'll add the Mormon church to this list. They believe they have apostolic succession directly from Peter, James, and John, as these 3 apostles appeared in a vision and ordained Joseph Smith and others.
Apostolic authority is the authority granted by apostolic succession, usually by some form of ordination.
According to Steven Lambert, ThD, apostolic authority has been lost
Now, the problem is, since the church fell into apostasy in the third and fourth centuries, it has not been recognizing these offices, and erroneous cessation theories purporting that the apostolic and prophetic offices ceased with the death of the Apostles of the Lamb have been prevalent, especially in mainline denominational doctrine. And then, somewhere along the line, primarily, as I said, within the last hundred years or so, local churches began calling the chief minister "the pastor," even though that office, according to Scripture, does not entail government, per se.