Busy Box is something that you install on your Android Device to give you some additional handy LINUX / UNIX based commands. You need Busy Box installed because some commands are not available to you and you made need them for some root level tasks. Here is some info on what Busy Box is.
BusyBox combines tiny versions of many common UNIX utilities into a single small executable. It provides replacements for most of the utilities you usually find in GNU fileutils, shellutils, etc. The utilities in BusyBox generally have fewer options than their full-featured GNU cousins; however, the options that are included provide the expected functionality and behave very much like their GNU counterparts. BusyBox provides a fairly complete environment for any small or embedded system.
Also in Titanium Backup if you hit the problems button, it will download and install busybox