The credit card companies don't charge interest - that would be the bank.   The bank also gets a cut of the processing company's (e.g., Visa's) "service charge"; one reference I've seen says that the issuing bank (i.e., the one where you have your account) gets half and the vendor's bank and Visa split the other half.

Credit card interest is basically interest on a short-term loan, and should be pretty competitive with other loans. For example, the credit card rate that I'd pay at my own bank - if I maintained a balance - is 9.9%. Home equity loans are 10.4% and personal loans 12%.