An acquirer is a company that specializes in processing payments. Through its network of accredited partners (or acquiring network), it enables a store to offer various payment conditions to its customers.
In exchange, the acquirer receives the payment information, processes it and passes it to the card brand (when the payment method is credit card) and the issuing bank.
For your store to be able to receive payments, you must enable its communication with an acquirer. This is done through a payment gateway, which must be configured to process the payment conditions. With that, purchases can follow the approval flow.
When everything is in order and a purchase is authorized by the other players, the acquirer is responsible for transferring the values (which the issuing bank receives from the customer) to the account of your store.