I'm also wondering what each of those does. Seems like with the Ecommerce solution I'm considering that everytime I think I've got it all figured out, it turns out that we need yet another service to make it work (Merchant Account, Gateway, now a Payment Processor)
Somewhat related to this question:
What is the difference between merchant account and payment gateway?
I would take a look at these two links, they explain the details of each player.
From those links below:
Payment gateway - Software used to transfer payment information from the merchant
to acquiring bank.
Credit card processors (or third-party processors) - Merchant services providers
that handle the details of processing credit card transactions between merchants,
issuing banks, and merchant account providers. Web site operators usually must first
establish their own merchant account before contracting for credit card processing
Merchant Account is not defined.