Basically an electronic payment gateway is the virtual connectivity between your website and the transaction processor (which might approve / decline the transaction) or send it over to the card association. The gateway is used in conjunction with a merchant account. (Hopefully others can give you a few names as an example - I am not allowed.)
A 3rd party processor - Internet Payment Service Provider (IPSP), you basically use their merchant account account to process credit card transactions. Usually consumers are directed to the IPSP's secure website to process the transaction.
With an electronic payment gateway, you can usually process credit card transactions on your secure website to maintain consistency when checking out