Apostolic Succession describes how the teaching and governing authority of the Church, established by Christ himself through his Apostles under the leadership of and in communion with Saint Peter, has been handed down through the centuries in the Catholic hierarchy.
When Christ established his Church as the one assembly of God's faithful people, he gave her a visible structure and hierarchy. He did this by selecting Twelve Apostles to serve as her first bishops, and from among these men he chose Saint Peter as their head, the "Rock" upon which he would build his Church. Christ "formed [his Apostles] after the manner of a college or a stable group, over which He placed Peter chosen from among them" (LG 19; cf. Luke 6:13; John 21:15-17). In giving St. Peter the "keys to the kingdom," Christ appointed his chief Apostle as his vicar, his source and sign of ecclesial unity on earth.
"Just as in the Gospel…Saint Peter and the other apostles constitute one apostolic college, so in a similar way the Roman Pontiff, the successor of Peter, and the bishops, the successors of the apostles, are joined together" (LG 22; cf. CIC 330). Saint Peter and the Apostles would teach in Christ's name under the guidance of the Holy Spirit, who guarantees that the Church's teaching authority (Magisterium) will always be protected against doctrinal error. (Cf. CCC 85-88,830-834,882, 936)
-The Didache Bible
Soon after Christ had ascended to Heaven, the Apostles recognized the need to appoint successors. They knew that the mission Christ entrusted to them was to continue beyond their deaths and endure until the Second Coming, so they elected Saint Matthias to take the place of Judas. Later, Christ himself called Saint Paul to be an Apostle. Saints Timothy and Titus were also among the first men designated and ordained to share in the apostolic mission. (Cf. CCC 77,883-896, 1087)
The early Fathers of the Church such as Pope Saint Clement I (d. ca. AD 99) testify to this succession (cf. I Clement, 44:1-3). All Catholic bishops today assume their office through episcopal consecration and trace their authority in an unbroken line to the original Apostles of first-century Christianity. The Pope himself is a direct successor to St. Peter. (Cf. CCC 1555-1560,1576)
-The Didache Bible
Peter, John, James, Andrew, Philip, Thomas, Bartholomew, Matthew, James son of Alphaeus, Simon the Zealot, Jude Thaddaeus, and Matthias.
All of the apostles except for Judas Iscariot spent the remainder of their lives preaching the Gospel.
According to Tradition, Saints Peter, Andrew, James, Philip, Bartholomew, Matthew, Thomas, James the son of Alphaeus, Simon the Zealot, and Judas Thaddeus were all martyred for the faith.
After His Resurrection, Jesus foretold Peter’s future martyrdom.
[Jesus said], "Amen, amen, I say to you, when you were younger, you used to dress yourself and go where you wanted; but when you grow old, you will stretch out your hands, and someone else will dress you and lead you where you do not want to go.” He said this signifying by what kind of death he would glorify God. John 21:18-19
Saint John, the Apostle who remained with Jesus at the Crucifixion, was the only Apostle to die a natural death – and this in spite of attempts to kill him by boiling him in oil.
Judas Iscariot, after betraying Jesus, committed suicide (see Matthew 5 and Acts of the Apostles 1:18).
In the above scripture passage, Matthew records the Apostles as the bearers of the obligations and the authority Christ committed to the Church. The text mentions only “the eleven,” that is, the Apostles minus Judas Iscariot. These are those whom Christ chose to take His place as pastors of His flock. This is the context of the command given to them to make disciples and to baptize, and of His promise to remain with them “to the end of the age.”
This also corresponds to Christ’s other commitments of authority and responsibility to the Apostles, such as the power of the keys (Matthew 16:13-18 and Matthew 18:8), and the forgiveness of sin (John 20:19-23). Since the Apostles all died, it also presumes that “to the end of the age” others will take their offices after them, as Matthias took that of Judas (Acts of the Apostles 1:15-26). Indeed, everywhere the Apostles went they appointed successors, as Paul appointed Timothy and Titus. In the 2nd century Saint Irenaeus of Lyon will list the successors of Peter in the Roman See, in contradistinction to the heretics of his day who can claim no such authority.
Christ is also with the Church “to the end of the age” in other ways, as well. However, He is uniquely and most importantly Present in the Holy Eucharist, by means of which His Holy Sacrifice is remembered and made Present, in the gift of Himself in Holy Communion, and by remaining with us in our tabernacles. In all these ways the continuity and unity of Christ’s redemptive ministry is guaranteed until He comes again (cf. John 16:13-15, John 17:2-21).
-from EWTN