It seems to appear mainly in cartoons. I always thought it was just a light to illuminate the inside of your mouth, nose, ear, etc. I can’t recall ever seeing a real doctor wear one.
I’ve seen real ones. Just a WAG but I think it’s to reflect light onto what the doctor is looking at.
It is (or at least was) used to perform an indirect laryngoscopy.
I believe the technical term is “Cranially-mounted reflective illumination device”
Head mirror or a headband mirror, apparently.
I’ve had it used on me. It is lowered in front of the doctor’s eye so he/she can see through the hole in the middle. The ring-shaped mirror is adjusted so that it reflects light from a nearby lamp and onto whatever the doctor is looking at, usually something deep inside a patient’s mouth (or other orifice, I suppose). It acts as a ring-shaped light and casts no shadows.
Just to put it all together in plain English: it’s a mirror. That is, a curved mirror with a hole in it. Doc swings it down over his eye, looks through the hole, and aims a light at the mirror. Light is reflected and concentrated on what Doc is looking at. And yes, I’ve been examined with one.
They are used mainly by otorhinolaryngologists. I haven’t seen one in a few years but I haven’t seen an otorhinolaryngologist in a few years, either. 
As of about 1980, nothing else would produce the shadowless light they need for direct examination. The alternative was/is an invasive procedure, unpleasant, and not without risk to the patient.
I used to use one, they’re very handy. They focus light in tight places. They also leave the doc’s hands free to work. As others have pointed out, ENT docs use them a lot to visualize throats, in conjunction with laryngeal mirrors. They’re also real nice for peering into earholes and nostrils. I miss mine!
But you really need a good backlight and a suitable chair to use it.