It differs by banks and countries, but e.g. my Dutch debit card only has an IBAN number on it, so anything set up to accept credit card numbers won't accept it.
Almost every online web shop here will accept debit cards, but it's through a Dutch specific payment mechanism (iDEAL).
That's because in the Netherlands most banks used Maestro (owned by Mastercard) and Vpay (owned by Visa) as international debit circuit.
The acceptance for Maestro and Vpay online is much lower than "normal" Mastercard Debit and Visa Debit. Acceptance for Mastercard Debit and Visa Debit is practically the same online as Mastercard Credit and Visa Credit.
Maestro is being sunset by Mastercard and no new card can be issued. Only Mastercard Debit. So the problem with the fact that Dutch debit cards (Maestro) are not accepted online will be solved in a couple of years.
Almost every online web shop here will accept debit cards, but it's through a Dutch specific payment mechanism (iDEAL).