A sleep doctor is a medical doctor who has additional special training in sleep medicine. Typically, the doctor is primarily trained in internal medicine, pulmonology or neurology and then completed additional training in sleep medicine. In order to be boarded (ie: certified or credentialed) in sleep medicine, the doctor must pass a rigorous exam (you can check the certification of a doctor on the American Board of Internal Medicine site here).
Once a doctor is boarded in sleep medicine, he or she can interpret sleep studies and may build a sleep medicine practice. Usually this doctor will see patients with other medical conditions too such as pulmonary or neurological conditions (based on her training).Alternatively, a physician may not obtain the sleep medicine credential required to interpret sleep studies, but has a special interest and pursues additional training in sleep medicine. These physicians can be primary care or internal medicine physicians, pulmonologists, psychiatrists, neurologists, ENTs or even dentists.