You can notify customers when a new invoice has been created using the New Invoice notification. This email will include a link to the invoice which will allow them to pay online if enabled or generate a PDF of the invoice to save for their records. You can customize the New Invoice template by following the instructions here: Customizing Notification Templates.
To send a new invoice notification, check the box to "Send New Invoice Email" when you create the invoice. The notification will only be sent when the invoice is approved and will not be sent if you save the invoice without approving.