There are a few major representations of the "oil". One, in the Old Testaments, it was instructed of Moses, by God, the use of the oil to anoint Aaron, the priest, for service for God. This was a great representation of separation and consecrating unto God to the priesthood. The anointing of God being represented on and in the presents of Aaron, which was also the same as the Holy Spirit in dwelling. Psalms 133:2 describes it as following: "It is like the precious ointment poured on the head, that ran down on the beard, even the beard of Aaron [the first high priest], that came down upon the collar and skirts of his garments [consecrating the whole body].
It also was the representation the Priests, Kings and Prophets that were anointed. Oil was poured on the head of the person being anointed. Kings were set apart through the ritual of anointing, which was performed by a prophet who acted in God's power and authority.
In the New Testament, anointing was represented with healing. The Holy Spirit's activities in a believer's life are picture in terms associated with anointing. The disciples anointed the sick in Mark 6:13 and James even instructed the elders to anoint the sick with oil (James 5:14).
Also in the New Testament, it represents the HOLY SPIRIT, which brings understanding to the believer (1 John 2:20, 27). The Holy Spirit anoints a person's heart and mind with the love and truth of God's word.