Let me take a shot at explaining it. I would describe it like this - it’s a programming language. It acts like the glue separating a variety of random commands from being a program. It’s commonly used to send the output of one program as input to another, while performing text manipulation in the middle as needed.
Bash “programs” are called scripts and technically Bash is a scripting language. Read more about Bash here. It is available on Linux, OSX, BSD, Solaris and other Unix-like operating systems.
A Video Introduction to Bash
I think a video will do a far better job of introducing you to Bash than I will. This is a introduction to some of the common tools available in a Bash shell on Linux and OSX.
Still confused? Leave a comment and I’ll try to point you in the right direction.