The New Testament tells us several ways through which we can share in God's gifts - "outward and visible signs of an inward and spiritual grace," to use a phrase from the Prayer Book catechism. So, for example, in baptism we are brought to new life in Jesus Christ, and in the Eucharist, we are bound together in that same life.
But here on earth, we carry God's gifts "in pots of clay" (2 Cor 4:7). The new life can still bring sickness, suffering and death. The sickness may be spiritual or physical - cured by forgiveness or by bodily healing, and we know from Jesus' work how closely linked the two forms of sickness are. (If you don't know, then read the story of the paralytic in Mark 2!).
In the case of spiritual sickness, the New Testament tells us to confess our sins to one another (James 5:16). Jesus called on his followers and gave them power to forgive these sins (John 20:23). And in the letter of James, this cure is extended to physical sickness. "Are any among you sick? They should call for the elders of the church and have them pray over them, anointing them with oil in the name of the Lord. The prayer of faith will save the sick, and the Lord will raise them up; and anyone who has committed sins will be forgiven." (5:14-15)
Here again, sickness and sin are closely linked. This does not mean that if we are sick, it is because we have sinned - although this can certainly be the case! It does mean that God can heal sickness, just as God can restore us from sin. It does not mean that God can do this by magic tricks - unexpected miracles, yes, but only miracles that are signs of God's love for the created world.
Why oil? Oil serves two functions. In the Old Testament, it consecrates - it sets aside for God's work. Jacob anoints the stone on which he dreamed his dream of the ladder. Moses anoints Aaron for his priestly work. Samuel anoints Saul as king. God's altar, God's priest, and God's king were pointers to Jesus, God's anointed - in Hebrew, Messiah, in Greek, Christ. (We still use oil to consecrate - at coronations and, more within our experience, at the baptism service. The oil we use is called chrism, and is a mixture of olive oil and balsam. Old writers saw these as signs of strength, richness and fragrance.)
Oil's second function was medicinal. Pouring oil on a wound prevented germs from reaching it or spreading. Oil-based perfumes could also slow down decay - which is why the dead were anointed for burial. And so we associate anointing (or unction, to give it its other name) especially with our ministry to the sick. This goes back to Jesus, whose disciples "anointed with oil many who were sick and cured them." (Mark 6:13)
In Western Europe, unction came to be used only for those in danger of death, through anointing the eyes, ears, nose, lips, hands and feet. In the 20th century, it became more directed at the healing of body and mind, at forgiveness and at spiritual strengthening. This is how we use it now, for now we are more aware that wholeness is a matter not only of our bodies, but of our souls - just as Mark's Gospel and James' letter continue to remind us.
HD