This will completely delete an email address and email account from the Mac Mail App, including all settings for the email account, and removing all associated emails from the Mail app in Mac OS X.
Pull down the Apple menu and choose ‘System Preferences’
Choose “Internet Accounts”
Select the email account you want to delete from the Mac from the list
With the email account selected, click the [-] minus button (or hit the Delete key on the keyboard)
Confirm that you want to remove the email account AND all of it’s associated emails and account settings from the Mac, including from the Mail application
The account will disappear from the list and all associated emails and settings from the email address will also vanish. You can repeat with other email accounts as necessary