Is a psychiatrist a doctor of medicine?

Yes, a licensed psychiatrist will have a medical degree such as the Doctor of Medicine (M.D.), Doctor of Osteopathic Medicine (D.O.), or The Bachelor of Medicine/Bachelor of Surgery (MBBS).

A bit more detail:

A psychiatrist has to go to medical school just as any other physician does, then has to study/work a few years longer doing his or her internship and residency to learn their chosen field of medicine in psychiatry. They treat the causes and symptoms of physical illnesses of the mind, such as Bipolar disorder, depression, schizophrenia, etc. and can write prescriptions just as any other physician can.

On the other hand, a psychologist, who treats emotional illnesses and disorders, is not a doctor of medicine and cannot write prescriptions. But once they get their doctorate degree (Ph.d.), they are a "doctor", just not a doctor of medicine, but a Ph.d. A very well known psychologist with a doctorate degree in psychology is Dr. Phil McGraw, more commonly known as Dr. Phil.