If you have set up your SQL Database Mail Operators and SMTP details correctly, but keep getting a message like this: "NOTE: Failed to notify 'operator' via email.", you may not have enabled the Alert System In the SQL Server Agent.
Here's how to enable the Mail System:
Right click "SQL Server Agent" in the Object Explorer and click "Properties"
Go to Alert System, and check the "Enable mail profile" box and select the correct Mail system and Mail profile.