Most of this site is dedicated to explaining how Mastercard and Visa credit card transaction work.  Mastercard and Visa accounted for $1.7 Trillion of the $2.1 Trillion in electronic card payments in the US in 2005 according to the Nilson Report.  The remainder are mainly store cards (without a Mastercard or Visa logo on them), Discover and American Express. 

Closed Networks Vs. Open Networks

Mastercard and VISA are known as open-loop systems because multiple banks participate in the system as issuers or acquirers (or both).  Discover and Amex are known as closed-loop systems because there is only 1 acquirer and issuer on each network, which is Discover or Amex themselves.  There really isn't any practical difference from the merchant's perspective except that there are a lot more VISA and Mastercards and the rates are more negotiable.

How to Accept Amex and Discover

Accepting American Express and Discover is easy and is usually facilitated by your Merchant Account Provider.  You just need to get an account number with Amex and Discover and tell it to your merchant account provider who will program it into your terminal or online processing gateway so that you can process Amex and Discover just as if they were a Visa or Mastercard.  

Costs of Amex and Discover

Discover discount rates are usually about 0.30% less than Visa and Mastercard and American Express is usually 1.50% higher than Visa and Mastercard.  Unlike VISA and Mastercard, you can't really negotiate with Discover and American Express, they don't discount (except for very large companies) and they don't do tricky things to gouge small and uninformed merchants.