From BiblePortal Wikipedia

Charles Buck Theological Dictionary [1]

In the primitive church, was an appellation given to all such churches as were founded by the apostles; and even to the bishops of those churches, as being the reputed successors of the apostles. These were confined to four, viz. Rome, Alexandria, Antioch, and Jerusalem. In after times, the other churches assumed the same quality, on account, principally, of the conformity of their doctrine with that of the churches which were apostolical by foundation, and because all bishops held themselves successors of the apostles, or acted in their dioceses with the authority of apostles. The first time the term apostolical is attributed to bishops, as such, is in a letter of Clovis to the council of Orleans, held in 511, though that king does not there expressly denominate them apostolical, but (apostolica sede dignissimi) highly worthy of the apostolical see. In 581, Guntram calls the bishops met at the council of Macon, apostolical pontiffs, apostolici pontifices. In progress of time, the bishop of Rome growing in power above the rest, and the three patriarchates of Alexandria, Antioch, and Jerusalem, falling into the hands of the Saracens, the title apostolical was restrained to the pope and his church alone; though some of the popes, and St. Gregory the Great, not contented to hold the title by this tenure, began at length to insist that it belonged to them by another and peculiar right, as being the successors of St. Peter. The country of Rheims, in 1049, declared that the pope was the sole apostolical primate of the universal church. And hence a great number of apostolicals; apostolical see, apostolical nuncio, apostolical notary, apostolical brief, apostolical chamber, apostolical vicar, &c.

Webster's Dictionary [2]

(1): (a.) Alt. of Apostolical

(2): (n.) A member of one of certain ascetic sects which at various times professed to imitate the practice of the apostles.