The differential, or final drive, is one of the last links in the chain before power is delivered to the wheels. The differential serves a key function, which is to prevent wheel hop during tight turns while still delivering the engine's power to the wheels. I'm not going to get into a long explanation of how differentials work, as we are mainly concerned with noises from the differential.
If your differential is making noise, it's likely it needs service. Outside of changing the fluid, there is little you can do. Like transfer cases, differentials can range from simple mechanical components to sophisticated engineered parts; it just depends on the vehicle and application. Even simple differentials can be difficult to set up without the proper tools, so if you find that you have a noise coming from your differential, it's best to leave it to a professional unless you have the tools and experience to tackle the job.
One thing to note on the fluid. You once again must use the correct fluid for best results. Some differentials are limited slip, meaning they deliver power to the wheels differently than a conventional differential. These differentials often require a special fluid or additive to be added to the oil for proper operation.
The funny thing is, these types of differentials are often the ones I find making noise. Much of that is due to improper service in the past. So heed the warning and use the correct fluid in your differential. It can save you a lot of money and headache later.