In CORE, you can add various email addresses in different screens such as Company Profile, Global Settings, User Profile and Employees to name just a few.
The email address and other details in the Company Profile screen displays on the header of your invoices. The email address set in Global Settings > Email Settings is a company-wide email address that CORE uses to send all outgoing emails, including the ones for invoices and reports. You can choose to override these settings and rather use the Email Settings at the user level (User Preferences) by checking the option Allow User Preferences to override these email settings. Check CORE Help Center for details.
When you set up a new user in CORE, you have the option to create a corresponding employee or vendor profile, and the email address of the user account is also copied to that profile. However, the email address gets associated with the user account and you cannot change it in the User Profile screen (main company menu > View Profile), as it is the email address that the user logs in to CORE with. You will have to set up a new user with the new email address, and delete the old one if you want to change the email ID of a user. Check out this article for details on how to correct the email address of a new user.
However, you can change the email address in the Employees or Vendors screen and also add more as part of the contact details. Check CORE Help Center for details.