Anointing of the Sick is one of the two sacraments of healing. The grace given in the sacrament is for strenth and healing - always spiritual healing and sometimes physical healing!
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
Any baptized Catholic can receive the sacrament of Anointing of the Sick for reasons of serious illness. Examples of appropriate times to receive Anointing of the Sick: a new diagnosis of a serious illness, a relapse in a serious illness, an elderly person whose frailty becomes more pronounced, or before a serious operation, etc.
The Anointing of the Sick is not just a sacrament for those only who are at the point of death...
If a sick person who received this anointing recovers his health, he can in the case of another grave illness receive this sacrament again. If during the same illness the person's condition becomes more serious, the sacrament may be repeated. It is fitting to receive the Anointing of the Sick just prior to a serious operation. The same holds for the elderly whose frailty becomes more pronounced.
- CCC 1514-1515
Last Rites is the term for the combination of three sacraments given to those who are dying: Anointing of the Sick, Confession (when possible), and Communion (Viaticum). Viaticum means "food for the journey," and is the term for Communion given to the dying.
Anointing of the Sick is a sacrament and, like the other sacraments, grace is objectively given to the person through the outward physical sign and action. The seven sacraments are insituted by Christ and entrusted to the Church for the faithful to receive grace. The grace of Anointing of the Sick is defined in Catechism 1520-1523 here.
Healing Nights are a place to receive prayer for a much broader array of healing needs. Grace is not objectively given in the way it is is a sacrament but we come before the Lord asking for his healing and strength. We pray for one another, as our Lord commanded, and entrust ourselves, and our needs to Him.