Lighting, Light Fixtures, Ceiling and Exhaust Fans - lighting fixture in bathroom

Hello there everyone. I need some help, please. I rented a place a few months ago, and the upstairs bathroom, the fixture has 2 lights, but only one was working, and that was plenty for me,as I dont much like alot of light. lol
Anyway, the other one stopped working, so I changed the bulb, and it fizzed out, right in front of me, I did this with several bulbs, the bulbs are good, cause I took them to another electrical source, and they work. I must have some sort of a short in the fixture, itself. Has anyone got any ideas as for where to start. I dont know much about electrical, but am very mechanical inclined. If someone could walk me thru it, I am sure I can find the problem and fix it myself, with some help from someone, thanks Ashley

Define "fizzed out".

Also, please identify what kind and wattage of bulbs the fixture takes.

Hello John and tks for responding.When I saying fizzing. the bulb kinda blinks a few times and then goes out. It takes standard watt bulbs. I looked at the one that has never worked since I move in here, where the bulb screws into the bottom of it is all black, must have been a short in that one also., tks

Could be
Bad socket on the fixture.
Bad wire connection to the socket.

Buy a new fixture and throw the old one away.

Hello there, and tks for the advice, that is my plan. I thank you always, you guys are always so helpful, have a great day!!!! Ashley