1. Click the File tab.

2. Click Options.

3. Select Mail and then uncheck all three boxes under Message arrival.

4. Click OK.

Now, we need to create rules to display notifications only from certain users.

5. Under the Home tab, click Rules and select "Manage rules and alerts." 

6. Click New Rule.

7. In the Rules Wizard, click "Play a sound when I get messages from someone." If you don't like sound alerts, just click "Display mail from someone in the New Item Alert Window."

8. Click "people or public group."

9. Select a name from your address book, or create a new one and then click From and OK.

Now, when this person emails you, you’ll get a notification. Others will remain silent. You’ll have to repeat the steps for anyone else you want to get notifications from.