Your fluorescent light is not working. The first thing to try is replacing all the fluorescent tubes in the fixture. If the light still doesn’t work, you will want to remove the inner cover that houses the ballast (the big black rectangle thing). Using a power tester and with the light switch on, carefully check for power coming into the light fixture.
If you have power in to the light fixture, you most likely have a bad ballast, or bad socket. Check for loose wiring and if everything look snug and there are no black marks on the sockets, most likely the ballast is shot.
Here is a great video showing you how to replace a ballast in a fluorescent light fixture.
