There are three emails that gets sent from the Payment System.  These can be found in the Invoices  --> Invoice Settings area of the Payment System dashboard. 

The Email & Thank You is the email content that is emailed out as a confirmation for a payment made. The content in the Thank You For Payment Content, is the page content the payer will see online after making the payment.

Date Due Email is the email sent when the invoice is due.  The invoice is created with a due date and this email is automatically sent out as the due date approaches, depending on the setting on the Email Days Before setting.

Quick Send Email is the email sent when the Quick Send Email icon (located below the invoice on the dashboard) is clicked.

Failed Payments

If you have recurring items that a client has subscribed to through the payment system and a recurring payment fails credit card processing, the system will send the Recur Failed Email associated with that item. The system will attempt to run the credit card two more times, each one 3 days from the last time before it stops trying and cancels the payment item.