Wikipedia has a lot to say on it:
en.wikipedia.org/wiki/Anointing_of_the_Sick
I've been to services for healing in my church, and have been anointed at them, and found it very comforting. Its not just confined to those who are near to death, or those who suffer from a physical problem, but also for those who are seeking mental comfort too.
What tends to happen is that people come forward in turn, apart from everyone else (so they are not overheard) can state what/who they want prayed for, are prayed over, and anointed. This is done as part of a service of prayer and healing.
Its all done rather gently and you are NOT expected to run from the church shouting "I'm healed I'm healed", nor is any implication made that if you aren't healed its because you don't have enough faith.
Is this the sort of thing you mean?