Why do you need an appointment of a health care representative in Connecticut?
If you become unable to handle your own medical care because of illness, an accident, or age, the right legal documents are your way of making sure your wishes are known. When you do not write down your wishes about the kinds of medical treatment you want (or don't want) to receive and name someone you trust to oversee your care, these decisions can be placed in the hands of doctors, estranged family members, or sometimes even a judge who may know very little about what you would prefer. You also should plan for the time that you are completely incapacitated with no hope of recovering from a fatal illness or fatal accident, as to what you wishes are about using life support or donating your organs or not. 
What are health care forms called in Connecticut?
There are three basic kinds of health care documents that everyone should make A Durable Power of Attorney, Appointment of a Health Care Agent, and a Living Will.
First, you'll need a document naming a trusted person to direct your health care if you are unable to do so yourself. In Connecticut, this form is called an Appointment of a health care Agent.
Second, you can create a document setting out the types of medical treatment you would or would not like to receive in certain situations. This document is often known as a Living Will. This document expresses your wishes if you are in a permanent vegetative state and your condition is irreversible and the only thing that can keep you alive is artificial means of life support. It also addresses your wishes with regard to whether you want to donate your organs or not. Connecticut calls this form a Document concerning health care and withholding or withdrawal of life support systems.
Third, you can create a durable power of attorney, providing one you trust to assume management of all of your financial affairs, medical decisions, etc. while you are still alive if you were to become mentally incapacitated temporarily or permanently. This document is null and void upon the death of the person who provides this durable power of attorney, at which time the will and trusts that were created by this deceased person will determine the disposition of all of the assets and liabilities of the decedent's estate.