A fixed rope is used when the rope does not need to be retrieved from the bottom of the rappel. Times when you would do this would be for caves or for rappelling activities.
To set up a fixed rappel on a single strand:
For a fixed rappel on double strand:
Fixed ropes are the safest way to rappel because there is virtually no chance of the rope coming unhooked from the anchor. If you're just rappelling single strand, you don't need as much rope as for a retrievable setup.
You have to unhook the rope when you are at the top of the cliff, not at the bottom.