Issuer is an institution from which the consumer gets his credit card. Usually it’s simply a bank (hence the popular term issuing bank), but it might also be e.g. a credit union. Such bank is associated with credit card brands, like Visa, MasterCard or American Express.
The issuing bank is actually the institution, which grants you the credit. So they are the ones, who check your credit history, debts. And later they decide what your credit limit is.