When a customer checks out on my shop, who do they pay?

Your customer will pay you directly, via the payment provider you have set up - you can configure these in your admin area: Modules > Payment.

Once the order has been confirmed, you will then need to submit this order in your dropship account (unless you have the direct order module).

Here is a list of our payment provider partners we recommend: http://www.dropshipping.co.uk/catalog/gateways.php