What is this sacrament?
 "The Sacrament of the Anointing of the Sick is given to those who are seriously ill by anointing them on the forehead and hands with duly blessed oil..." (Lumen Gentium - Second Vatican Council)  
 
Who may receive the Anointing?
 In the Catholic Church this sacrament was occasionally called "Extreme Unction" and associated with "Last Rites" because people often waited until the moment of death to receive it.  The sacrament of Anointing of the Sick is not only given to those at the point of death.  It can be administered to anyone struggling with a serious illness to give them strength and healing they need.  It is not regularly repeated unless one's condition worsens, one falls into a different serious illness, or, after recovering, falls seriously ill yet again.  
 
What does the Anointing do?
 The healing that occurs in this sacrament of anointing is not necessarily physical healing. The sacrament is given for spiritual healing and grace is given to the recipient so that they may find strength, peace and courage that are needed in the face of serious illness or the frailty of old age. (CCC 1520)  In old age and serious infirmity one can easily find their souls sick or weak, easily discouraged, downcast, despairing, and in need of spiritual healing.  This sacrament helps us against these temptations of the evil one.
In the bible, James also indicates that this sacrament brings "forgiveness of sins."  This has been understood as forgiveness of venial sins, since the Sacrament of Confession is the ordinary way in which our mortal sins are forgiven.  There is no doubt that physical healing can arise through this sacrament, as has happened many times, but physcial healing is not the primary purpose of this sacrament.
 
When should I call?
Do not wait until the moment of death or until somebody has died to request this sacrament. Sacraments are for the living and are no longer helpful to those who have died.  In an emergency, try to call a Catholic priest, any Catholic priest, as soon as possible.