Shell is a command interpreter. It is an middle layer between the operating system kernel and the user, it's also a fairly powerful programming language. A shell program is called a script, is an easy-to-use tool for building applications by "gluing together" system calls, tools, utilities, and compiled binaries. Virtually the entire repertoire of UNIX commands, utilities, and tools is available for invocation by a shell script. If that were not enough, internal shell commands, such as testing and loop constructs, lend additional power and flexibility to scripts. Shell scripts are especially well suited for administrative system tasks and other routine repetitive tasks not requiring the bells and whistles of a full-blown tightly structured programming language. Scripts are very usable in scheduling jobs and for repetitive tasks.
According to Wikipedia,
Bash is a Unix shell written by Brian Fox for the GNU Project as a free software replacement for the Bourne shell (sh). Released in 1989, it has been distributed widely as the shell for the GNU operating system and as the default shell on Linux and Mac OS X. It has been ported to Microsoft Windows and distributed with Cygwin and MinGW, to DOS by the DJGPP project, to Novell NetWare and to Android via various terminal emulation applications.
Bash is a command processor, typically run in a text window, allowing the user to type commands which cause actions. Bash can also read commands from a file, called a script. Like all Unix shells, it supports filename wildcarding, piping, here documents, command substitution, variables and control structures for condition-testing and iteration. The keywords, syntax and other basic features of the language were all copied from sh. Other features, e.g., history, were copied from csh and ksh. Bash is a POSIX shell but with a number of extensions.
The name itself is an acronym, a pun, and a description. As an acronym, it stands for Bourne-again shell, referring to its objective as a free replacement for the Bourne shell. As a pun, it expressed that objective in a phrase that sounds similar to born again, a term for spiritual rebirth. The name is also descriptive of what it did, bashing together the features of sh, csh and ksh.