Disclaimer: IANAP. Please notify me of any errors or omissions.
Thanks to paraclete for some appropriate clarifications.
Psychosis is commonly defined as an illness where the patient experiences an altered subjective experience of reality and no insight into this condition.
The key symptoms of psychosis are hallucinations, delusions and thought disorder.
A hallucination is a sensory experience that arises from the mind, without an external cause. Hallucinations are most often auditory, e.g. a voice or several voices commenting (often in a persecutory fashion) on the person's thoughts or actions.
Delusions, on the other hand, are firmly held beliefs that are out of keeping with the patient's social and cultural background (thus strong belief in a member of a religious community is not necessarily "delusional"). See the node at delusion for their classification. The form of the delusions varies among various types of psychotic states. For example, paranoid schizophrenics usually suffer from delusions of reference, persecution or monitoring, whereas those undergoing manic episodes may enjoy subjectively pleasant delusions of grandeur.
Thought disorder refers to disorganized forms of thinking that are difficult or impossible to follow for most people. Psychotics may, for example, seem to jump from one subject to another, following a thread of thought that is indiscernible for others. Psychotics often make use of neologisms (new words) to express their thought. Thought disorder is usually more serious in chronic, institutionalized psychotics.
Psychotic episodes often involve sudden changes in emotional states that seem strange to others. The psychotic may have bouts of "undue happiness" (..) as well as hyperactivity. He or she may laugh or become angry at odd times. Social withdrawal is quite common.
Schizophrenia is a form of psychotic illness, but schizophrenics do not suffer from psychosis all the time. Other causes for psychosis include drugs of abuse (e.g. amphetamine psychosis), drug withdrawal (e.g. delirium tremens), infections, brain tumors, metabolic abnormalities, nutritional deficiencies, and dementia. Mania and depression sometimes involve psychosis. Also, extremely stressful events may trigger (usually temporary) psychoses. In addition, psychedelic drugs induce states of mind that can be remarkably similar to certain psychoses. (I believe Stanislav Grof recommended that psychologists and psychiatrists take LSD to induce a model psychosis in themselves to better emphatize with their psychotic patients.)
There is no single cause for psychotic symptoms. Most forms of psychosis are neurochemical in origin and some are due to physical injury or illness, but sometimes a person merely predisposed towards psychosis will assume and support another's delusions in exceptional circumstances. This is known as a shared psychotic disorder or folie à deux.