What is busybox
Busybox allows you or programs to perform actions on your phone using Linux (copied from Unix) commands. Android is basically a specialized Linux OS with a Java compatible (Dalvik) machine for running programs. The Android kernel is a modified version of the Linux kernel (that is why the Android kernel must always be open source). Busybox gives functionality to your phone that it does not have without it. Many programs, especially root programs such as Titanium Backup, require busybox to perform the functions of the program. Without busybox installed your phone is much more limited in what it can do.