What is apostolic succession, and did the early Church teach it?

Clement of Rome fl. c. 96 ADFirst Letter to the Corinthians 44.1–3

Our apostles also knew through our Lord Jesus Christ that there would be strife over the office of bishop. For this reason, having received perfect foreknowledge, they appointed those already mentioned, and afterwards gave a further rule, that when these fell asleep, other tested men should succeed to their ministry. Those, then, appointed by them, or later by other reputable men with the consent of the whole church, who have ministered blamelessly to the flock of Christ ... we think it unjust to remove from their ministry.

The first statement of succession: the apostles appointed leaders and arranged for others to succeed them. Rome writes to restore deposed presbyters at Corinth.

Hegesippus (quoted by Eusebius) c. 110–c. 180 ADMemoirs, quoted in Eusebius, Church History 4.22.3

When I came to Rome I made a succession list down to Anicetus, whose deacon was Eleutherus. Soter succeeded Anicetus, and after him Eleutherus. And in each succession and in each city things are as the law preaches, and the prophets, and the Lord.

Succession here proves that the same teaching was kept in every city.

Irenaeus of Lyons (quoted by Eusebius) c. 130–c. 202 ADAgainst Heresies, quoted in Eusebius, Church History 3.3.3

The blessed apostles, having founded and built up the church, handed over the ministry of the episcopate to Linus. Paul mentions this Linus in his letters to Timothy. Anencletus succeeded him, and after him, in the third place from the apostles, Clement received the episcopate, who had seen the blessed apostles and spoken with them, and still had their preaching ringing in his ears and their tradition before his eyes.

Irenaeus lists twelve bishops of Rome to Eleutherus, against Gnostics who claimed a secret tradition.

Irenaeus of Lyons c. 130–c. 202 ADAgainst Heresies 4.26.2

So we must obey the presbyters who are in the Church, those who hold the succession from the apostles, as we have shown, who with the succession of the episcopate have received the sure gift of truth, according to the Father's good pleasure.

Irenaeus goes on that those who break away from the principal succession are to be held suspect, as heretics, schismatics, or the proud. This part of Against Heresies survives in its ancient Latin translation, not in Greek.

Tertullian c. 155–c. 220 ADPrescription against Heretics 32.1–2

Let them produce the origins of their churches; let them unroll the list of their bishops, running down in succession from the beginning, so that their first bishop had as his author and predecessor one of the apostles or of apostolic men who stayed with the apostles. For this is how the apostolic churches show their records: as the church of the Smyrnaeans reports Polycarp placed there by John, as the church of the Romans reports Clement ordained by Peter.

Tertullian adds that churches founded later are apostolic if they share the same faith (32.6).

Cyprian of Carthage c. 200–258 ADLetter 66, to Florentius Pupianus 66.8

The bishop is in the Church and the Church is in the bishop, and anyone who is not with the bishop is not in the Church. ... The Church, which is catholic and one, is not cut or divided.

Cyprian's most quoted sentence on the episcopate, written to a layman drawn toward the Novatianist schism: those who creep about in secret, without peace with God's priests, flatter themselves that they are in communion.