Confirmation is a sacrament distinct from Baptism, but it has a similar function. It completes the work of Baptism and gives a person the fullness of the Holy Spirit to be steadfast in their faith. By receiving this sacrament, a person is ready to stand up for and defend their faith.
In the Bible – Institution
In His Last Supper Discourse, Jesus speaks of the imminent coming of the Holy Spirit to the Apostles (John 14 – 17) and the impact He will have on them. In the Acts of the Apostles 2, the Holy Spirit does come to them at Pentecost and the transformation is astounding. Previously, they had been cowering in the upper room for fear of the Jews, but with the coming of the Holy Spirit, they are not afraid to go out and preach the gospel. Peter preaches that day, is ridiculed as drunk, and yet continues to do so bravely. As time goes on, the apostles risk imprisonment and even death to continue preaching Jesus, and this change begins on the day of Pentecost, when they receive the Holy Spirit.
In the Bible – Sacrament in Action
There are several places in the New Testament, particularly Acts of the Apostles, in which the Apostles administer the Holy Spirit in the sacrament of Confirmation, and it is clearly a different sacrament than baptism. “The two went down and prayed for them that they might receive the Holy Spirit (for as yet the Spirit had not come upon any of them; they had only been baptized in the name of the Lord Jesus). Then Peter and John laid their hands on them, and they received the Holy Spirit." - Acts 8: 15 – 17. This instance demonstrates that the special coming of the Holy Spirit is different than baptism, for it records that they had already been baptized. The apostles needed to be present in order to lay hands on the faithful so that they could receive the Holy Spirit. "On hearing this, they were baptized in the name of the Lord Jesus. When Paul had laid his hands on them, the Holy Spirit came upon them, and they spoke in tongues and prophesied." - Acts 19: 5 – 6. Again, the coming of the Holy Spirit through the apostle and the laying on of hands is distinct from Baptism, and these particular people received the gift of tongues after they received the Holy Spirit in Confirmation.
Early Church Fathers
The bishop, imposing his hand on them, shall make an invocation, saying, ‘O Lord God, who made them worthy of the remission of sins through the Holy Spirit’s washing unto rebirth, send into them your grace so that they may serve you according to your will, for there is glory to you, to the Father and the Son with the Holy Spirit, in the holy Church, both now and through the ages of ages. Amen.’ Then, pouring the consecrated oil into his hand and imposing it on the head of the baptized, he shall say, ‘I anoint you with holy oil in the Lord, the Father Almighty, and Christ Jesus and the Holy Spirit.’ Signing them on the forehead, he shall kiss them and say, ‘The Lord be with you.’ He that has been signed shall say, ‘And with your spirit.’ Thus shall he do to each. - Hippolytus The Apostolic Tradition 21–22
Some say in regard to those who were baptized in Samaria that when the apostles Peter and John came there only hands were imposed on them so that they might receive the Holy Spirit, and that they were not re-baptized. But we see, dearest brother, that this situation in no way pertains to the present case. Those in Samaria who had believed had believed in the true faith, and it was by the deacon Philip, whom those same apostles had sent there, that they had been baptized inside—in the Church...Since, then, they had already received a legitimate and ecclesiastical baptism, it was not necessary to baptize them again. Rather, that only which was lacking was done by Peter and John. The prayer having been made over them and hands having been imposed upon them, the Holy Spirit was invoked and was poured out upon them. This is even now the practice among us, so that those who are baptized in the Church then are brought to the prelates of the Church; through our prayer and the imposition of hands, they receive the Holy Spirit and are perfected with the seal of the Lord. - Cyprian Letters 73[72]:9
But beware of supposing this to be plait ointment. For as the Bread of the Eucharist. after the invocation of the Holy Ghost, is mere bread no longer, but the Body of Christ, so also this holy ointment is no more simple ointment, nor so to say common, after invocation, but it is Christ's gift of grace, and, by the advent of the Holy Ghost, is made fit to impart His Divine Nature. Which ointment is symbolically applied to thy forehead and thy other senses; and while thy body is anointed with the visible ointment, thy soul is sanctified by the Holy and life-giving Spirit. - Cyril of Jerusalem, Catechetical Lectures (On Chrism) 21:3
And then remember that you received the seal of the Spirit; the spirit of wisdom and understanding, the spirit of counsel and strength, the spirit of knowledge and godliness, and the spirit of holy fear, and preserved what you received. God the Father sealed you, Christ the Lord strengthened you, and gave the earnest of the Spirit in your heart, as you have learned in the lesson from the Apostle - Ambrose On the Mysteries 7:42
The living water of holy Baptism is given to us as if in rain, and the Bread of Life as if in wheat, and the Blood as if in wine. In Addition to this there is also the use of oil, reckoned as perfecting those who have been justified in Christ through holy baptism. - Cyril of Alexandria Commentary on the Minor Prophets 32
These things have I written unto you concerning them which seduce you; that ye may know that ye have an unction, and the unction which we have received from Him may abide in you." In the unction we have the sacramental sign [of a thing unseen], the virtue itself is invisible; the invisible unction is the Holy Ghost. - Augustine Homily 3 on the First Epistle of John par. 12
These authors clearly recognize that Confirmation is an important sacrament and that it is distinct from Baptism. Through the administration of this sacrament, the Holy Spirit comes to those who receive it and gives His gifts of grace to them.