Apple accepts credit cards, but debit cards have been phased out as an acceptable payment method starting over year or so ago. They continued to work for many people for a long time, but once they don't, you must then use a credit card or other form of payment.
Until December 6 last year, Apple's Terms and Conditions specifically omitted any mention of debit cards, and specifically included credit cards as an acceptable form of payment. Now, they are silent on the issue and only say you must use "an acceptable payment method". To determine what that is, you have to research more:
When you make a purchase in the iTunes Store, Mac App Store, App Store, or iBookstore your account will be billed at the time of or shortly after your transaction. If you use a credit card or PayPal account for a transaction, Apple may obtain preapproval for an amount up to the amount of the order. ...
All fees will be billed to the credit card or PayPal account you designate during the registration process.
No mention of debit cards.
I hope this eliminates any uncertainty regarding this problem.