Acquirer is the financial institution licensed by card networks like Visa and Mastercard to process card transactions on behalf of a merchant. It is the acquirer that sends the transaction to the card network, receives the approval or decline response from the card's issuing bank, and credits the approved amount to the recipient.
A payment gateway can be connected to more than one acquirer at the same time. This routing between different acquirers is what allows a second route to be tried when the first one declines a transaction, a practice known as multi-acquirer routing.