How do I replace a broken light fixture?
In hindsight, I think this past question might have also helped out:
How can I change the bulb in this three-clawed, ceiling-mounted dome light fixture?

A picture of what you currently have would allow us to help you out more, but I'll go on the assumption that it needs to be replaced..

In this case, it's a pretty simple process. Switch off the power back and the breaker and finish removing the dome if you haven't yet, again the above question link could help. If you're able, test the sockets with some form of voltage tester (or even a good bulb that works on another lamp) to ensure that the power is off. Then on the base that is mounted to the ceiling, there will either be some screws holding it up or a center threaded rod which will have a nut and washer holding the fixture to the ceiling. In either situation, unscrew it, but keep a hold on base as it will only be held up by the wires after it has been disconnected from the support. You should be able to pull it down enough to see the wiring above the base within it's electrical box. Disconnect the light from the wiring and you're good to go. Follow the instructions that are included with your replacement fixture and you'll have light back in no time.