Shopify will allow you to void and refund payments, restock your inventory, and even send a cancellation email to your customer all with the click of a button. No more fussing around with payment gateways!

Open the order in your admin, and click on "Cancel this Order" in the upper right.

When you click the order cancellation button, you’ll be presented with a window similar to the one below:

Using this function will also automatically dispute your Shopify transaction fees.  If the order was placed within your current billing cycle, the fees will be removed from your upcoming invoice.  If they were on a previous billing cycle, you will be issued transaction fee credit on your next invoice.
 

Can I do partial refunds?

Yes. You're going to do the same steps as above but refund less, according to what exactly was returned. Any unfulfilled products will automatically be restocked.

If you want to send a special email notification with partial refunds, you can click on "Contact Customer".