What is the equivalent, in a car, for the term “light fixture” in a structure such as a house or apartment, mounted on a ceiling, that a light bulb is screwed into?
Last night when I returned to my car, I noticed that the rear “dome” light was on. The car started all right, but I found that there was nothing I could do to get the light to go out when I closed a rear door (I tried both), so I unplugged the entire unit so I could drive home and leave the car parked in the driveway at home overnight without running down the battery. I wanted to explain the situation to a mechanic today, but I didn’t know what to call the device that the light fits into. 
Lamp socket … I guess … you’ve explained your problem perfectly here … I’m sure your mechanic will know what to do …
Check the dome light switch in your vehicle. I sometimes bump mine with my knee when I get in my truck turning the dome light on. The switch will have 3 positions, always on, on when doors open, always off. It took me an hour to discover the problem the first time I turned on the dome light and it wouldn’t go off.
Check what racer72 said, and be aware that sometimes the dome light control is integrated with your regular headlight switch. Sharing your year and model will help with diagnosis.
If it is not something as simple as a switch in the wrong position, it could well be a failure of one of your door switches. Certain brands/models have a bad reputation for this. I guess what we are saying is that the failure is almost certainly not the dome light itself, but in a switch that controls it.
Hmmmm…actually this is a Mercedes-Benz, with a cluster of buttons in the front near the front interior light. I’ll test those…
rear dome light, rear cabin light, rear ceiling light …
The ones on the roof are dome lights so they spread the light everywhere.
Ones above seats or the dashboard tend to be flat or lensed for direction … spot lights…
Just call it the dome light or inside light. They will know what you are talking about. And read your owner’s manual as for the operation of the light.