For some years I've worked without the support of any bug tracking system (BT for short) but now it's hard to do without. I think that its standard usage is clear enough, otherwise Wikipedia has a good explanatory page. In this post I want to cover some more things you may find it useful for.
Your Historical Memory
Sometimes bugs tend to represent over time. In this case they are called regressions. A good starting place when you want to create regression tests is your bug tracker. All the bugs with steps to reproduce and support files are stored in a single place. The QA Team just needs to take the information from there and put them into the automatic test suite (because you use it, right?).
Besides, sometimes you choose to not fix some bugs for a variety of reasons. If there is some sort of workaround to get rid of them, it must be registered in your BT for future reference.
Not Only For Bugs
Nobody said that you cannot use it for keeping track of the development of new features. Many BTs provide separated categories for bugs and new implementations. In this way it's easier to filter them and make statistics to be shown to your manager.
Decent Bug Trackers give you the possibility to set the priority for each entry. Using this feature, you can easily manage the work of your team.
And why not using it also for something unrelated to programming, such as the documentation?
Make It Public
Warning! This suggestion should be carefully evaluated by the highest levels of your company because it can be potentially dangerous.
Generally speaking, letting your users signal bugs and requiring new features may increase the level of loyalty. On the other hand, your company must be well organized in order to quickly answer every request in a proper manner. And sometime the proper manner may be "we won't do that". Of course, too many "no" may give a bad idea of your company to potential customers.
Another issue is that one of your customers may feel like you are taking more care in resolving issues signaled by other customers.
However, if your company agrees on this point, it's better to have separated bug trackers for users and developers.
A bug tracking system is one of the most useful things that your company can adopt to manage defects and the project workflow.
Image by Messer Woland from Wikimedia Commons licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.