I'd like to be notified whenever comments are made on the site which require administrator approval before publishing; basically, I'd like to know when something new appears in the Approval Queue. I can't see any obvious settings for this.
Sounds like something Rules should be able to do.
Check out the Comment Mail module -- if you're going to get an email each time there is a comment you might as well simplify the process by adding approval functionality:
The Comment Mail module allows an email to be sent to the site administrator(s) when new comments are posted. A link in the email allows quick approval, editing, deletion, and/or banning of the poster's IP address.
Word of caution: I've used this method before but quickly turned it off after the spam comments broke my spirit. Make sure you've got a robust anti-spam strategy setup (ex. antispam, mollom, captcha/recaptcha, etc.)