A Medical Proxy is a person who is specifically appointed to make medical decisions and give informed consent for a patient who is unable to do so, either due to the fact they're unconscious, are a minor, or are otherwise mentally incompetent.
In most cases, a Medical Proxy is not necessary for such persons, as there is a clear line of medical decision making power for persons not competent to act on their own behalf. For example, one spouse has the clear authority to speak for the other spouse, and parents have the clear authority for their children. In most other cases, a relative of a person has the authority.
However, in many situations, the closest relative is not the appropriate person to act for the patient. In those cases, the patient or a court may appoint a person to act as their medical decision maker.
The use of Medical Proxies in House has been very extensive: In Three Stories, an ailing House gave Stacy medical proxy for him while he was in an induced coma. She used the power to go against House's express wishes.
In Euphoria (Part 2), Foreman appointed Cameron to stand up for him against House when his father would not. Cameron insisted on a dangerous but potentially life saving procedure.
In Finding Judas, a judge appoints Cuddy as a patient's medical proxy.