I believe that Visa and MasterCard are the most popular and widely accepted. Other credit cards are sometimes not accepted at certain vendors, because the vendor has to pay a fee in order to be able to accept certain credit cards. Since Visa and MasterCard are the most popular, most vendors opt to not pay that fee for American Express.
The most universally accepted credit card other than the Visa card is known as the Master card. With this card you are allowed to make various transactions in your account regardless of your location in the world.
Master card and Discover are accepted most places as well as Visa. The reason a lot of places don't take American Express is the fee they charge retailers is higher than the other companies. A percentage of every sale that is made using a credit or debit card goes to that companies card and American Express charges more on that fee.