The laying on of hands
What the apostles' laying on of hands bestows, and whether it is a distinct rite from baptism.
Acts 8:14-17
14Now when the apostles who were at Jerusalem heard that Samaria had received the word of God, they sent Peter and John to them,
15who, when they had come down, prayed for them, that they might receive the Holy Spirit;
16for as yet he had fallen on none of them. They had only been baptized in the name of Christ Jesus.
17Then they laid their hands on them, and they received the Holy Spirit.
What each tradition says
Rabbinic / Midrashic
Bacher and Lauterbach suggest the custom may have declined because Christianity adopted laying on hands to dedicate disciples as independent officiants and transfer the teacher's office. The Apostles laid hands while praying on the seven Jerusalem disciples (Acts vi. 6); the ceremony is mentioned three times in Timothy as dedicating a teacher of the Law and recipient of divine grace.
Wilhelm Bacher, Jacob Zallel Lauterbach, Ordination (The Jewish Encyclopedia, 1901-1906) — Section 'Confined to the Patriarch' (the Apostles' adoption of the laying on of hands, Acts vi. 6)
Patristic
Chrysostom explains that great signs had been done, yet they had not received the Spirit. They had received the Spirit of remission of sins, but not the Spirit of miracles.
John Chrysostom, Saint Chrysostom: Homilies on the Acts of the Apostles and the Epistle to the Romans (NPNF1-11; Schaff, ed.) — Homily XVIII (heading 'Acts VII. 54'), on Acts 8:14-17
Catholic
Aquinas holds that the Holy Ghost is given to the baptized for strength, as to the apostles at Pentecost and by the apostles' imposition of hands in Acts 8:17. Since the Holy Ghost is not given except with sanctifying grace, it is evident that sanctifying grace is bestowed in this sacrament.
Thomas Aquinas, Summa Theologiae, Tertia Pars (Fathers of the English Dominican Province trans., Second and Revised Edition, 1920) — Tertia Pars, Q. 72 (The sacrament of Confirmation), art. 7, 'I answer that'
Orthodox
Cyril of Jerusalem says that after you had come up from the pool of the sacred streams, there was given an Unction. This Unction is the anti-type of Christ's anointing, and it is the Holy Ghost.
Cyril of Jerusalem, The Catechetical Lectures of S. Cyril, Archbishop of Jerusalem (NPNF2-07; Schaff, ed.) — Lecture XXI (Mystagogical Catechesis III, 'On Chrism'), § 1
Reformed / Protestant
Calvin says they had only been baptized: they had received the grace of common adoption and regeneration offered to all the godly in baptism. The gifts of the Spirit given to certain people were extraordinary, serving to set forth the kingdom of Christ and the glory of the gospel.
John Calvin, Commentary upon the Acts of the Apostles, Volume 1 (Fetherstone trans.; Beveridge ed.) — Comment on Acts 8:16 (lemma 'They were only baptized'), within the comment on vv. 14-17
Evangelical
Barnes argues this passage should not be used to support the rite of confirmation. He says there are now no apostles, and what is spoken of here is entirely different from confirmation. It was to confer the extraordinary power of working miracles, while confirmation is for a different purpose.
Albert Barnes, Albert Barnes' Bible Commentary — Note on Acts 8:15 (on the gift of the Holy Ghost and the 'rite of confirmation')