In a Living Will you appoint the person you wish to make end of life medical and life sustaining decisions for you if you become incapable of doing so. These rights apply, for instance, to someone who is on life support or suffering from a terminal disease or condition. 
Consider the unfortunate circumstances of Terri Schiavo which could have been avoided with a Living Will. In Schiavo’s case, there was nothing in writing to tell her family what she wanted when heart failure caused her to fall into a vegetative state. Schiavo was only 26 years old when this occurred in 1990. A lengthy legal battle ensued in Schiavo’s home state of Florida involving its court system, legislature, and Governor Jeb Bush. Schiavo passed away on March 31, 2005. If Terry Schiavo had expressed her wishes in a written document, her loved ones could have avoided the entirely ugly and expensive court process. Many years earlier, Karen Anne Quinlan, a young college student, even younger, ingested drugs and alcohol and entered a vegetative state. Without any direction from her, she was kept alive in that vegetative state for many years without being able to see or acknowledge her surroundings.
If you have specific wishes regarding similar situations, then a Living Will is your only assurance that your wishes will be carried out.
A Living Will, formally called an Advanced Medical Directive, specifically states whether or not someone wishes to initiate or discontinue life support and address other end of life decisions. This document directs an agent to carry out the wishes of the writer. A Living Will gives you control over unforeseen circumstances and peace of mind for you and your family that your wishes will be carried out.