Bash (bash) is one of many available (yet the most commonly used) Unix shells. Bash stands for "Bourne Again SHell", and is a replacement/improvement of the original Bourne shell (sh).

Shell scripting is scripting in any shell, whereas Bash scripting is scripting specifically for Bash. In practice, however, "shell script" and "bash script" are often used interchangeably, unless the shell in question is not Bash.

EDIT: Actually, the default scripting shell in Ubuntu is dash, while the default interactive shell (what you get if you pull up a terminal) is Bash. Nonetheless, the two terms are still mostly interchangeable.