Usually, you don't have to do anything. Your card on file is charged automatically after the auction closes.
Your invoice shows Paid once the charge goes through. Paying online doesn't change your pickup dates. You still need to pick up during the pickup window.
Larger invoices don't cancel your purchase. Take care of payment before you head to pickup unless the auction allows payment onsite. Wire payment instructions should be included with your invoice.
Future invoices will be charged to your new default card.
Banks sometimes give a clear reason, like Insufficient funds. Other times, they give a general one, like Do not honor.
Common causes are a debit card's daily spending limit or an expired card on file. To update your card, go to Profile, then Billing.
A declined payment doesn't cancel your purchase. You still need to pay before you pick up.