Three terms used interchangeably and incorrectly across the industry. They describe three different layers: the gateway encrypts and routes the card data; the merchant account is the bank account that holds the settled funds; the processor (acquirer) clears the transaction through the card networks. Stripe / PayPal / Square bundle all three.
The software layer that takes card data from the checkout, encrypts it, tokenises it, and routes the authorisation request. Authorize.net and NMI sell only the gateway and pair with a separate merchant account / processor.
The merchant account is a bank account that holds settled card funds. The processor (acquirer) is the entity that has card-network membership and actually clears the transaction. Chase, Elavon, Fiserv, Worldpay, Global Payments are processors; Stripe and PayPal are also processors but bundle the gateway.