A payment acquirer is the bank or licensed payment institution that enables a merchant to accept card and other electronic payments. It signs the merchant up, routes its transactions into the card networks, pays out the settled funds and is answerable to the card scheme for the merchant's conduct. The role is explained in full in the entry on the Acquirer.
The term is used loosely in the industry. Strictly, the acquirer is the institution holding the scheme licence, often an acquiring bank, while processors, gateways and a payment facilitator may handle the technical work or onboard sub-merchants under the acquirer's licence. The merchant's funds arrive through a merchant account. On the other side of every card payment sits the card issuer, which serves the cardholder.