Acquirer processor is another name for an acquiring processor, the company that handles the technical work of card acceptance on behalf of an acquirer, and that entry covers the role in full.
The acquirer processor receives transactions from merchants' terminals, websites and gateways, sends authorisation requests to the card networks, builds the clearing files, calculates what each merchant is owed and produces settlement and reporting data. Some acquiring banks run this processing in-house, while many outsource it to specialist processors so that they can focus on sales, underwriting and risk.
The two terms mean the same thing; "acquirer processor" stresses that the firm works for the acquirer, while "acquiring processor" stresses the activity. Both are distinct from an issuer processor, which provides the same kind of service to card issuers on the other side of the transaction, and from a payment gateway, which mainly connects online merchants to processors.