Voiding a payment effectively deletes a payment from the order.  A payment can only be voided if the date of the payment is the current date, except for a credit card payment that was successfully processed by your payment provider.  Your credit card payment provider does not support voiding same-day transactions, nor does it support same-day refunds.  Your only option in this case is to refund the credit card payment the next day or later.

If the payment can be voided, a "Void" link will appear.  Click it and a dialog will ask you to confirm.  Once confirmed, the payment will disappear from the order.

Voiding Scheduled Payments

A scheduled payment or post-dated payment, such as with cheques, cannot be voided, but you can cancel them by clicking its "Cancel" link.  See Cancelling Pending Payments for more information.

Voiding Past Payments

You cannot void a payment with a date in the past.  If the payment is in the past and is "Successful", you will have a refund button so you can  refund the payment.  

If your past payment is "Pending", for example, when a post-dated cheque payment's date has passed and the "Payment has been received" hasn't been checked yet, you may only mark the payment as being received and then refund the payment.