I work in the industry so maybe I can shed some light:

1. Visa and MasterCard are both associations of banks that give license to their members to allow them to issue the cards and use the association acceptance networks.

2. MasterCard recently became a publicly traded company while Visa is still fully owned by the member banks. The member banks are large shareholders and have voting rights in the publicly traded MasterCard

3. The acceptance rates on a global basis for both Visa and MC are about the same. However, as others have pointed out those acceptance rates may vary by region. MasterCard is making a push in Europe where Visa has tended to have an advantage. MasterCard has an advantage in most of Latin America. There really is little to no difference in the US. Moral - have one of each since there is little difference

4. The associations often provide marketing or sponsorship dollars to large organisations or companies for exlcusive acceptance rights - ie. the NFL and the Olympics only accept Visa.

5. The same can be said for certain banks who tend to issue a majority of their cards with one or the other association

6. AMEX is a separate acceptance network which is wholly owned and controlled by AMEX. Same for the old Diners (before the deal with MC) and Discover.