A delusion is the belief by a patient about a fact that is not true. This is often a symptom of some type of brain damage, but is common in many types of mental illness as well. It differs from a hallucination in that it happens when the patient is otherwise lucid and that patient can often determine that hallucinations are not true. It differs from a psychotic break in that the patient can rationally discuss the delusion.
In Honeymoon, House realized that Mark Warner was suffering from delusions when he realized that Mark had not honeymooned in Paris like he said, but that his brain scan at the time showed that he truly believed he had. This led House to the correct diagnosis - acute intermittent porphyria.
In Guardian Angels, the patient, in addition to hallucinating that her mother was in the room with her, was under the apprehension that her mother was alive. Although the hallucinations continued during the first stages of treatment, the delusion stopped right away and the patient accepted that it was a hallucination.