Bash scripts are scripts that run under the bash shell. A linux shell is a command line interpreter, that allows you to interface with a linux system. When you login to a linux system with SSH, you will be presented with a shell.
The default for most linux distributions is bash, which is one of many shells you can choose from. There is the default built-in shell called “sh” then there’s zsh, tcsh and others.
There is a popular framework out there for developers called Oh My Zsh which is a framework for zsh to make it awesome.