If an invoice is showing "Unpaid" that means that a payment or credit adjustment has not been applied or fully applied to this invoice. To learn how to apply existing payments click here, and an existing credit adjustments click here.
Applying a payment has no effect over the totals of a client, it mainly just changes an invoice status from saying Unpaid to Paid. There are some benefits to applying payments as far as Reporting goes. Invoice AR Aging, Payments Applied, Collections, and Invoice report all require payments to be applied to Invoices in order to show proper totals on these reports. There are other reports like Account Activity, Account Balance, Payments Received, etc., that do not require you to apply payments to invoices in order to run properly.