By default Bugzilla sends emails for a number of different reasons.

As a user you can control the kinds of emails you receive by navigating to Preferences / Email Preferences.

There you will see a grid email types and triggers. Select those that make sense for you.

As an administrator you have a few more areas of control.

To completely disable email for a specific user navigate to Administration / Users and find the user. Then check the Bugmail Disabled checkbox and click Save Changes.

To disable the “Your Bugzilla bug list needs attention” emails you need to turn off Whining. Navigate to Administration / Parameters / Email and scroll down to whinedays. Set the value to 0 and click Save Changes.