If you use offline payment method on your site, when users make payment using that payment method, the status of the payment method is Pending. When you receive payment (for example via bank transfer), you need to publish/ approve the payment record:
When you do that, the system will change status of the payment record to Paid. Also, it will send payment approved email to users. You might want to go to Payment Form -> Configuration, look at Messages tab, adjust the following message items if needed: