Just looking the fixture itself, and giving the probability of other points of failure... I would guess maybe the ballast on the lamp has given out (probably the silver box in the image). A ballast steps up the voltage so the lamp operate correctly, and they fail on occasion. Finding the replacement ballast might be the hard part and sometimes expensive. It might be easier to just get a new fixture and use CFLs (which have built-in ballasts).
At the same time purchasing a new fixture and replacing it will also confirm, in a relatively easy way, that the wiring to your fixture is working. Just be sure to throw the breaker to that fixture. Don't depend on someone wiring the light switch correctly and given your uncertainty of the status of your home wiring.