To understand apostolic succession, you must go back to the early Church Fathers. Here is what they wrote:
"Through Our Lord Jesus Christ our Apostles knew that there would be strife over the office of episcopacy. Accordingly, since they had obtained a perfect foreknowledge of this, they appointed those men already mentioned. And they afterwards gave instructions that when those men would fall asleep, other approved men should succeed them in their ministry. Therefore, we are of the opinion that those appointed by the Apostles, or afterwards by other acclaimed men, with the consent of the whole church, and who have blamelessly served the flock of Christ in a humble, peaceable, and disinterested spirit, and have for a long time possessed the good opinion of all, cannot be justly dismissed from the ministry." St. Clement Of Rome ("The Epistle Of Clement To The Corinthians," c. 96 A.D.)
"In this order, and by this succession, the ecclesiastical tradition from the Apostles, and the preaching of the truth, have come down to us. And this is most abundant proof that there is one and the same life-giving faith, which has been preserved in the church from the Apostles until now, and handed down in truth." St. Irenaeus ("Against All Heresies," c. 180 A.D.)
"It is necessary to obey the presbyters who are in the Church - those who, as I have shown, possess the succession from the Apostles. For those presbyters, together with the succession of the bishops, have received the certain gift of truth, according to the good pleasure of the Father. But we should hold in suspicion others who depart from the primitive succession and assemble themselves together in any place whatsoever. For they are either heretics or perverse minds, or else they are schismatics who are puffed up and self-pleasing.... Therefore, it behooves us to keep aloof from all such persons and to adhere to those who, as I have already observed, hold the doctrine of the Apostles." St. Irenaeus ("Against All Heresies," c. 180 A.D.)
"He cannot be reckoned as a bishop who succeeds no one. For he has despised the evangelical and apostolic traditions, springing from himself. For he who has not been ordained in the Church can neither have nor hold to the Church in any way.... How can he be esteemed a pastor, who succeeds to no one, but begins from himself? For the true shepherd remains and presides over the Church of God by successive ordination. Therefore, the other one becomes a stranger and a profane person, an enemy of the Lord's peace." St. Cyprian Of Carthage ("Letter To Magnus," c. 250 A.D.)
In ( Against Heresies, Book 3, Chap. 3, before aprox. 195 A.D. ) Saint Irenaeus, Bishop of Lyons, wrote: " . . . The blessed Apostles, then, having founded and built up the Church, committed into the hands of Linus the office of episcopate. Of this Linus, Paul makes mention in the Epistles to Timothy. To him succeeded Anacletus; and after him, in the third place from the Apostles, Clement was allotted the bishopric. This man, as he had seen the blessed Apostles, and had been conversant with them, might be said to have the preaching of the Apostles still echoing [ in his ears ], and their traditions before his eyes. Nor was he alone [ in this ], for there were many still remaining who had received instructions from the Apostles. In the time of this Clement, no small dissension having occurred among the brethren at Corinth, the Church in Rome dispatched a most powerful letter to the Corinthians, exhorting them to peace, renewing their faith, and declaring the tradition which it had lately received from the Apostles . . . "
WHO did the Apostles appoint as the Bishop of Rome? Where was Peter?
One source is Here
There are hundreds more if you are interested in history rather than boilerplate apologetics.
BTW, are we drifting pretty far off the original topic?