BusyBox is an app that's not so useful in itself but essential in order to use a lot of other apps that require you to have a rooted phone.
Basically, what BusyBox gives you is a set of UNIX tools that will run in the background every time they're needed. In other words, the ideal is to have BusyBox installed on your phone without worrying too much about it so it can work for you every time it's needed (like when installing an application that requires root access).
BusyBox is kind of like a Swiss Army knife for Android. Even though doesn't do anything in particular alone, it will give you access to many benefits from third-party apps.
