Ash is a simple unix shell, as compared to bash, which is the more typical fully featured shell preferred on Linux systems.

Busybox is a program that implements an ash like shell, as well as a number of other common unix programs, though in a stripped down version, all in a single program. It is intended to provide a semi-typical unix like environment in a very space limited environment, hence, why it implements many ( stripped down ) programs as single program, and then has the separate program commands symlinked to the busybox binary, which figures out which program it is supposed to emulate and does its best to behave in a similar way to the fully featured program that it is replacing.

Busybox was originally developed for use in very space limited environments such as a single floppy boot disk, and continues to see use today in space limited environments like embedded systems such as dd-wrt.