Closed invoices are any invoices where the balance has reached $0. This includes any invoices where the payment amounts that were charged or recorded totaled up to the invoice total or any invoices where a write off amount was added to bring the balance on the invoice down to $0.
When an invoice is in closed status, you cannot enter any more payments or make changes to any line items. In order to do this, you'd have to reopen the invoice using the More Actions menu in the top right:
