Most payment gateways have a setting 'AuthorizedOnly'. If checked, this means that only customers whose accounts have the 'Approve for special payments' box checked will see this payment option at checkout. This is most commonly used for payment methods you may want to restrict to trusted customers, such as 'PO_OfflinePayment' (where a customer can order goods without a credit card and the invoice is sent to them for payment later).

If the 'AuthorizedOnly' box for a payment method is not checked, then this payment option is available to all users.

If the 'AuthorizedOnly' box for a payment method is checked, but the user is not approved for special payments, they will not see this option at checkout.