Dallin H. Oaks gave a talk in April 2010 called "Healing the Sick". It encompasses what I believe about blessings for the sick. He said:
There are five parts to the use of priesthood authority to bless the sick: (1) the anointing, (2) the sealing of the anointing, (3) faith, (4) the words of the blessing, and (5) the will of the Lord.Here is my short summary of them
1- The Old and New Testament spoke of anointing with olive oil. It is a symbol of purity and power.
2- Dallin H. Oaks said, "To seal something means to affirm it, to make it binding for its intended purpose." Usually another man seals the anointing after the first anoints.
3- Both the faith of those blessing and those being blessed are required. The faith of those being blessed is more important.
4- The words of the blessing are ideally in perfect harmony with the Lord's will. But it is not necessary in order for the Lord to bless.
5- The Lord knows best what will help us to live the happiest lives. Sometimes our will is not the Lord's. Trust in the Lord can bring us peace even when things happen that we do not want to happen.
My father in law mentioned how he talked about how we use the Melchizedek priesthood to give blessings for the sick. Hebrews 6:20 and Hebrews 7:1-2 talk about the Melchizedek priesthood. We believe that Jesus held this priesthood. That he gave it to his twelve apostles and that Peter, James and John restored this priesthood to Joseph Smith. And that it has been passed down since that time to us.