Euthanasia or assisted suicide describes a situation where a physician either administers a drug that will result in the death of a patient or provides the patient with access to drugs that are fatal in order to end the patient's life prematurely.  Doctor's often resort to it when the patient is in intense pain and is suffering from a terminal illness.

The episode which deals most directly with the subject is Informed Consent, where the patient, who is suffering from an undiagnosed illness, insists on being put to death rather than undergo a series of increasingly invasive tests.  Robert Chase appears to be the doctor who is most comfortable with the concept.  Gregory House is not opposed to the concept, but will only administer fatal drugs to a patient who he is sure is terminally ill, not a patient who is merely undiagnosed and suffering.  Eric Foreman is opposed to it, but does not take active steps to prevent other doctors from doing it.  Allison Cameron was also opposed to it, but was the person who eventually gave the fatal dose to the patient once the diagnosis showed that the patient could not be treated.

James Wilson faces the problem the most often as cancer patients often suffer terribly in the late stages of their disease.  In Known Unknowns he planned to deliver a speech in support of it, together with a description of his own experiences.  However, House realized that this would end Wilson's career and instead drugged Wilson and delivered the speech himself in the guise of Dr. Perlmutter

Euthanasia at Wikipedia