What triggers customer notification email to be sent?

Customer notifications for a product variant are sent once the inventory quantity is set to 1 unit or more. 

Products that are hidden (unpublished) won't trigger customer notifications to be sent (if the product later becomes visible the notifications will be sent).

You can set the inventory quantity by editing the product using your store admin. Back in Stock will be notified of the change and send the email automatically.

You can control number of emails sent or use a higher quantity than 1 to trigger notifications by setting your account delivery policy. See  Controlling email delivery for more details.