In the email templates there you can find the template of email called Affiliate - Invoice and Affiliate - On payout.
The email Affiliate - On payout is sent always when you label some commission of an affiliate as paid. The affiliate will get notification that his commissions has been paid.
If you want to send also the invoice, you need to specify its format in Configuration>Invoice format. When you label a commissions as paid, check the checkbox "Send generated invoices to affiliate" - it means, that hte email Affiliate - Invoice would be sent to affiliates.