Here is a way to have Drupal send you an email when a new comment has been posted on your site. This method uses the built in modules, Actions and Triggers, that come with Drupal core. You may want to check out my post on good contributed modules to add to Drupal to block spam, or else you will eventually receive more emails than you probably want to.
Log in and make sure the Actions and Trigger modules are enabled.
Navigate to the Actions administrative page
From the drop-down menu "Choose an advanced action", select "Send email..."
Provide a description, an email address, a subject for the email and your message. You can use string replacement tokens to make your emails easier to read, such as the following:
In this message, the email will tell you what page a comment was posted on, it will give you a link to the page and also give the default URL to go to the comment administrative page.
You can save this action and now you need to set up a trigger to define when this email will be sent.
Go to the Triggers admin page
Select the Comment tab
Under the "Trigger: After saving a new comment" fieldset, assign the action you just made from the drop down menu.