This is the list of invoices you created for the client. If the invoice is overdue, the Due date is marked in orange. Once the invoice is printed, you shouldn't edit it anymore.
To mark an invoice as paid, right-click on the specific invoice. Please also have a look at all other options, when right clicking. They are all self explanatory.
In this dialog you can also find the common filter and grouping option, known from the project and timestamp list. You can save you settings at any time.