Historically payment processing services were supplied by banks that were members of the Visa or Mastercard networks. Typically banks both issued credit cards and helped merchants process them, but over time the industry consolidated. Now, very few banks issue credit cards and the industry is dominated by a few large issuers – Elavon, Citibank, Capital One, MBNA / Bank of America, and Chase. Even fewer banks process credit cards. Banks found that it was not within their skillset to convince every small merchant to accept credit cards, and they began to outsource the selling of such services to small companies called ISOs (Independent Sales Organizations). They also found that massive scale helped reduce the cost of processing credit cards, so they began to outsource processing to a few giant processors including Elavon, TSYS, First Data and MSI.
Now, while some merchants buy their payment processing services directly from a bank, more typically they get their payment processing services from an ISO, which is responsible for selling the service to the merchant, providing technical support, processing the transaction (authorizing it and submitting it to the Visa or Mastercard network), bears the risk of chargeback(s), and sets the price of the services. If the ISO is a small ISO it might outsource some of those services (for example providing technical support) to a larger ISO. Except for the few very largest ISOs, the ISO outsources the actual processing to a larger company that focuses solely on processing.
While each of the firms discussed above provides merchant accounts and could legitimately be called the merchant account provider, in practice the phrase usually refers to whichever organization has responsibility for directly maintaining the relationship with the merchant.