From the Catholic Encyclopedia:
Chrism is used in the administration of the Sacraments of Baptism, Confirmation, and Holy Orders, in the consecration of churches, chalices, patens, altars, and altar-stones, and in the solemn blessing of bells and baptismal water. The head of the newly-baptized is anointed with chrism, the forehead of the person confirmed, the head and hands of a bishop at his consecration, and the hands of a priest at his ordination. So are the walls of churches, which are solemnly consecrated, anointed with the same holy oil, and the parts of the sacred vessels used in the Mass which come in contact with the Sacred Species, as the paten and chalice. If it be asked why chrism has been thus introduced into the functions of the church liturgy, a reason is found in its special fitness for this purpose by reason of its symbolical significance. For olive-oil, being of its own nature rich, diffusive, and abiding, is fitted to represent the copious outpouring of sacramental grace, while balsam, which gives forth most agreeable and fragrant odours, typifies the innate sweetness of Christian virtue. Oil also gives strength and suppleness to the limbs, while balsam preserves from corruption. Thus anointing with chrism aptly signifies that fulness of grace and spiritual strength by which we are enabled to resist the contagion of sin and produce the sweet flowers of virtue. "For we are the good odour of Christ unto God" (2 Corinthians 2:15).
Each year on Holy Thursday (that is, the Thursday before Easter Sunday), the Chrism Mass is held and the bishops combine sweet smelling balsam with olive oil, consecrate the chrism, and then separate it into many different containers for the priests in their diocese to use throughout the next year. There is more about this beautiful and symbolic mass here.
In scripture we see many examples of anointing with oil, especially for the sick and to consecrate people, especially priests and kings, and things to be used in the service of God. Here is a very small sampling of some of those scriptures, there were so many that I nearly got overwhelmed trying to decide which to use!
With this sacred anointing oil you shall anoint the meeting tent and the ark of the commandments, the table and all its appurtenances, the lampstand and its appurtenances, the altar of incense and the altar of holocausts with all its appurtenances, and the laver with its base.
When you have consecrated them, they shall be most sacred; whatever touches them shall be sacred. Aaron and his sons you shall also anoint and consecrate as my priests. (Exodus 30:26-30)
They drove out many demons, and they anointed with oil many who were sick and cured them. (Mark 6:13)
Is anyone among you sick? He should summon the presbyters of the church, and they should pray over him and anoint (him) with oil in the name of the Lord, and the prayer of faith will save the sick person, and the Lord will raise him up. If he has committed any sins, he will be forgiven. (James 5:14-15)
Mary took a liter of costly perfumed oil made from genuine aromatic nard and anointed the feet of Jesus and dried them with her hair; the house was filled with the fragrance of the oil. (John 12:3)
I'll end with this lovely prayer from the Chrism Mass link above.
"God our maker, source of all growth in holiness, accept the joyful thanks and praise we offer in the name of your Church. From that time forward, through the sign of holy chrism, you dispense your life and love to men. By anointing them with the Spirit, you strengthen all who have been reborn in baptism. Through that anointing you transform them into the likeness of Christ your son and give them a share in his royal, priestly, and prophetic work."