A psychosomatic illness is one with physical manifestations but without any known or identifiable physical cause. On a basic level, virtually everyone will scratch their head if you tell them a story about headlice, and a large number will feel nauseated or worried about themselves (what was in that last meal I ate?) when hearing about someone else's food poisoning experience.
The mind has a powerful influence over the body, and it is quite possible for a person to feel extremely physically ill without any real physical reason, especially if they suffer from an anxiety disorder. Worrying over the physical symptoms exacerbates them, leading the patient to the doctor for reassurance that there isn't actually anything seriously wrong with their health. A psychosomatic illness will generally improve or disappear after this reassurance is delivered, e.g., by negative test results.
Some doctors automatically dismiss the symptoms of any patient who has been diagnosed with an anxiety disorder on these grounds without any further treatment other than verbal reassurance. If the illness is genuinely psychosomatic this may be sufficient, but it can also be a very dangerous thing to do as many serious illnesses, such as heart disease and cancer, have initial vague symptoms.
Many now-identifiable illnesses have previously been labelled as 'hysterical' or 'psychosomatic', for example asthma, allergies, and migraines. Some illnesses are under debate, including Multiple Chemical Sensitivity and Gulf War Syndrome.