Bony fishes have an effective pumping system that involves the mouth and the outer cover of the gills, called the operculum. When the fish's mouth opens, the operculum closes, drawing water into the mouth. When the fish closes his mouth, the operculum opens and allows fresh water to cross the gills. Oxygen passes through the thin wall of the gills and into the blood. Other fish, such as the tuna, have less effective pumping systems and must swim constantly to keep fresh oxygenated water flowing over their gills. They usually swim with their mouths partly open.
To learn more, read How Do Fish Breathe.