BASH is the GNU Bourne Again SHell, the successor to the classic Unix Bourne sh (shell). It's the default shell for various Linux/GNU systems.
There are a variety of interpreters that receive commands either interactively or as a sequence of commands from a file. The Bourne-again shell (Bash) is one such interpreter. Bash implements the standard Bourne Shell (sh), and offers numerous additions.
From the Free Software Foundation's Bash page:
Bash is an sh-compatible shell that incorporates useful features from the KornShell (ksh) and C shell (csh). It is intended to conform to the IEEE POSIX P1003.2/ISO 9945.2 Shell and Tools standard. It offers functional improvements over sh for both programming and interactive use. In addition, most sh scripts can be run by Bash without modification.
Read the Bash manual for technical details.
Books and Resources
Additional reading materials include: