Descent of the Holy Ghost by Gustave Doré 1865
Pope St. Pius X explains Confirmation in his Catechism:
Confirmation is the sacrament which gives us the Holy Ghost, imprints on our souls the mark of a soldier of Jesus Christ, and makes us perfect Christians by confirming us in the faith and perfecting the other virtues and gifts received in Baptism.
The seven gifts of the Holy Ghost are: Wisdom, Understanding, Counsel, Fortitude, Knowledge, Piety, and the Fear of the Lord.
The matter of this sacrament, besides the imposition of hands by the bishop, is the anointing of the forehead of the baptised with sacred chrism; and for this reason it is also called the sacrament of Chrism, that is Anointing.
Sacred Chrism is oil of olive mingled with balsam, and consecrated by the bishop on Holy Thursday. In this sacrament the oil, which is unctuous and strengthening, signifies the abounding grace which is diffused over the soul of the Christian to confirm him in his faith; and the balsam, which is fragrant and prevents corruption, signifies that the Christian, strengthened by this grace, is enabled to give forth a good odour of Christian virtue and preserve himself from the corruption of vice.
The form of the sacrament of Confirmation is this: “I sign thee with the Sign of the Cross, and I confirm thee with the chrism of salvation, in the Name of the Father, and of the Son, and of the Holy Ghost. Amen.”
In administering the sacrament of Confirmation the bishop first stretches his hands over those to be confirmed, and invokes the Holy Ghost upon them; next, he anoints the forehead of each one with sacred chrism in the form of a cross, saying the words of the form; then he gives each one confirmed a light stroke on the cheek with his right hand, saying: Peace be with you; finally, he solemnly blesses all those he has confirmed.
The anointing is made on the forehead, where signs of fear and shame appear, in order that he who is confirmed may understand that he should not blush at the name and profession of a Christian, nor fear the enemies of his faith.
A light stroke is given to the person confirmed to show him that he should be ready to bear all insults and endure all sufferings for the faith of Jesus Christ.
We find this great sacrament in the Holy Scriptures several times:
Now when the apostles, who were in Jerusalem, had heard that Samaria had received the word of God, they sent unto them Peter and John. Who, when they were come, prayed for them, that they might receive the Holy Ghost. For he was not as yet come upon any of them; but they were only baptized in the name of the Lord Jesus. Then they laid their hands upon them, and they received the Holy Ghost. (Acts 8:14-17)
And it came to pass, while Apollo was at Corinth, that Paul having passed through the upper coasts, came to Ephesus, and found certain disciples. And he said to them: Have you received the Holy Ghost since ye believed? But they said to him: We have not so much as heard whether there be a Holy Ghost. And he said: In what then were you baptized? Who said: In John’s baptism. Then Paul said: John baptized the people with the baptism of penance, saying: That they should believe in him who was to come after him, that is to say, in Jesus. Having heard these things, they were baptized in the name of the Lord Jesus. And when Paul had imposed his hands on them, the Holy Ghost came upon them, and they spoke with tongues and prophesied. (Acts19:1-6)
Wherefore leaving the word of the beginning of Christ, let us go on to things more perfect, not laying again the foundation of penance from dead works, and of faith towards God, Of the doctrine of baptisms, and imposition of hands, and of the resurrection of the dead, and of eternal judgment. (Heb 6:1-2)
In whom you also, after you had heard the word of truth, (the gospel of your salvation;) in whom also believing, you were signed with the holy Spirit of promise,…(Eph 1:13)
The following are three early Church Fathers writing on Confirmation. More can be found with Tertullian, St. Hippolytus of Rome, St. Ephraim, St. Cyril of Jerusalem, etc.
St. Theophilus of Antioch writing in 181 AD “To Autolycus”:
Are you willing to be anointed with the oil of God? It is on this account that we are Christians: because we are anointed with the oil of God.
Pope St. Cornelius wrote in 251 AD to Fabius of Antioch:
As [Novation] seemed about to die, he received Baptism in the bed where he lay, by pouring – if indeed, such a man can be said to have received it at all. And when he recovered from his illness he did not receive the other things which, in accord with the law of the Church, it is necessary to have; nor was he sealed with by the bishop. And since this was not done, how could he have the Holy Spirit?
St. Cyprian of Carthage to Jubalianus, Bishop of Maurentania 254 AD:
Some, however, 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, however, re-baptized. But we see, dearest brother, that this situation in no way pertains to the present case. For those in Samaria who had believed, had believed in the true faith; and it was by the deacon Philip, who those same Apostles had sent there, that they had been baptized inside, in the Church, which is one, and in which alone it is permitted to give the grace of Baptism and to absolve sins.
For the reason, then, that they had already received legitimate and ecclesiastical Baptism, it was not necessary to baptize them again. Rather, that only which was lacking was done by Peter and John; and thus, prayer having been make 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 are then brought to the prelates of the Church; and through our prayer and the imposition of hands, they receive the Holy Spirit and are perfected with the seal of the Lord.
Speray's Catholicism in a Nutshell 