You need a credit card, preferably your own.
From the App Store Frequently Asked Questions (FAQ)
Accepted payment types
Shopping the App Store requires an iTunes Store account. When you create an iTunes Store account, you must provide a payment method for your purchase. The available payment methods vary by country, but typically you can use credit cards, gift cards, store credit, or PayPal.
The iTunes Store Terms and Conditions also mentions credit cards only, not debit cards.