A merchant account is a type of bank account that allows businesses to accept payment by debit or credit cards.
A payment gateway facilitates the secure transfer of information between you and your merchant account.
These work together to form the basic "plumbing" that allows you, the merchant, to accept and process credit card transactions and have the money collected end up in your bank.
Chargify is a complete management system on top of this... kind of like the automated control system that turns on the water to your lawn and garden.
If you'd like to learn everything you'll ever want to know about merchant accounts & payment gateways, and the easy ways to get these things, here is a Complete Description by one of our founders.