Your parent shell will be running the whole time. That will be sh. Any other processes spawned by that shell will also be running. Try pstree to show parent-child relationships.
BTW, if you use bash-specific features that aren't in POSIX Bourne shell, you should use #!/bin/bash, not #!/bin/sh. Some systems have bash, but have a lighter-weight /bin/sh.
I am very new to bash scripting... sorry if this is a dumb question.
Not dumb. Basic, but only once you understand how Unix processes work, (and how whatever you're using in OS X that shows you "service" names, since that's not a word that would make sense in any Unix context in this situation.) So you're dealing with a fair amount of complexity, and I don't blame you for asking.
Maybe OS X looks at process group leaders or something to come up with a "service name", if that's what it really calls it. I think that would be the process name of whatever process is running in the foreground (i.e. that you didn't fork off with & at the end of it, so the shell is waiting for it before executing the next command.)