From how stuff works The oldest sharks, the modern sharks' ancestors, didn't have to constantly swim to breathe. Rather, they all pumped water through their mouth and over their gills. This method is known as buccal pumping, named for the buccal, or cheek, muscles that pull the water into the mouth and over the gills. Many sharks retain this method today, such as nurse sharks, angel sharks and carpet sharks, also known as wobbegongs. Skates and rays, the shark's cousins, also breathe this way. These species tend to spend most of their time lying on the bottom of the ocean floor.
Some sharks, however, have completely lost the ability to breathe by buccal pumping, and these are the sharks that will indeed drown if they stop swimming and ramming water. These sharks are known as obligate ram breathers (or obligate ram ventilators); only about two dozen of the 400 identified shark species are required to maintain this forward swimming motion [source: Bennetta]. These include the great white shark, the mako shark, the salmon shark and the whale shark.