This very much depends on the country. In some places there are debit card networks which are neither MasterCard nor Visa nor Maestro. In such situations your bank's debit card may actually be useless for online payments. Even if the cards are Maestro and an online retailer happens to accept Maestro (somewhat rare in my experience), that may not help you since banks will often blanket deny any card-not-present transactions.

Tell that to my bank lol.