Explanation (One True Faith)
The Anointing of the Sick
See Catechism §§ 1499–1532. The Holy See’s text
What it is
The Anointing of the Sick is the Sacrament for a baptised person in danger from illness or old age. It is not reserved for the last five minutes of life, and it is not a substitute for a physician. It is Christ present to a body that is failing.
In Scripture
James writes: Is any man sick among you? Let him bring in the priests of the Church, and let them pray over him, anointing him with oil in the name of the Lord. The prayer of faith shall save the sick man, and if he be in sins, they shall be forgiven him. The Apostles had already anointed the sick during the public ministry.
What the Church teaches
Only a Priest or a Bishop may anoint. The Sacrament may be repeated if the person recovers and falls ill again, or if the same illness becomes more grave. It is not given to a person who is already dead. The Church prays for the dead. The Church does not anoint them.
See Catechism §§ 1499–1532.
How it is celebrated
The Priest lays on hands and anoints the forehead and the hands with the oil of the sick, blessed by the Bishop, ordinarily, on Holy Thursday. If the person can, the rite is joined to Confession and to Viaticum, the last Communion for the journey.
What it does
The Sacrament gives grace to bear the illness, the forgiveness of sins if the person could not confess, and sometimes the restoration of health, if that is for the soul’s good. It also prepares the person for the passage into eternal life. A cure, when it comes, is a gift. The Sacrament has not failed when the person dies in grace.
If this is your question
Call the Priest before the person can no longer speak, if you can. Do not wait for the family to agree that death is likely. A sick person who asks is already reason enough. Lourdes itself sends people to doctors. So should you.