In the alphabet soup of providers of psychotherapy and mental health care, psychiatrists have the most stature and, consequently, the greatest influence on the practice of mental health care and what defines mental illness. Before we elaborate, allow us to make clear just who a psychiatrist is:
Psychiatrists are medical doctors. This means they have completed medical school, a residency in psychiatry, and (in the United States) are board-certified and licensed to practice psychiatry. This practice includes psychotherapy as well as the diagnosis of mental illness, the prescribing of psychotropic medications (those medications used in the treatment of mental disturbances ranging from depression to schizophrenia) along with a handful of other medical procedures.
There is much overlap with other professionals in regards to what psychiatrists are allowed to do. Many professionals can make mental health diagnoses and practice psychotherapy. Psychiatric nurse practitioners (and in some states, psychologists) can prescribe some psychotropic medications and, in fact, most psychotropic medications are prescribed by primary care doctors.
All of that said, psychiatrists are generally considered the authority on all of these matters. It is the American Psychiatric Association, for example, that published the (THE) manual of mental disorders, the DSM-V, which names and defines the diagnostic criteria for all mental health diagnoses.