.sh files are to be interpreted by a "vanilla" Unix shell and should contain NO commands that need a certain shell (ksh, Bash, . . .) in order to run them properly.
It had become habit to tag bash scripts with the extension ".sh". Wrong! So many scripters who use bash-specific commands or syntax (or don't wish to make special efforts to see that they're not doing so) began using the extension ".bash" instead. Actually, I'd say that's an improvement.
So: If you see "myscript.bash", run it with "bash myscript.bash". You might want to look through it first though, to see if there's anything in it that you wouldn't feel comfortable running from the command line. That's what a ".bash" is, after all. A set of commands for the command line of Linux's most common user shell.
The files caitlyn mentioned are startup files bash looks for by name. Both are often referred to and are quite important -- they'll affect how Bash works.