Verizon, Sprint, etc use CDMA. It's a quasi-proprietary protocol by Qualcomm, and CDMA phones don't use SIM cards.
GSM phones use SIM cards. (AT&T and T-Mobile, mainly, in the U.S.)
Carriers don't design phones. They customize and subsidize phones designed by cell phone companies like Nokia and Motorola (and Apple, tee-hee). Most cell phone manufacturers make all sorts of phones, both CDMA and GSM, and arrange to sell them through the appropriate carriers.