When you breathe in air, the air travels from your nose, down your trachea (windpipe), and into your lungs.
As the lungs branch into smaller and smaller airways, the end in specialized sacs called alveolae. Here, oxygen passes through the lung membranes into the bloodstream, and waste products like carbon dioxide flow out of the blood and into the air, and are subsequently expelled when you breathe out.
Fish also need oxygen to live, but their lungs are not designed to extract oxygen from the air.
Instead, by passing the water through their specialized organs (called gills), they can remove the oxygen and eliminate waste gases.
Since humans do not have gills, we cannot extract oxygen from water. Some marine mammals, like whales and dolphins, do live in water, but they don't breathe it. They have developed a mechanism to hold their breath for long periods of time underwater. Eventually, however, they have to come to the surface to exhale and then take a new breath.
This answer was provide by Dr. Beth Ann Ditkoff. This and other interesting answers can be found in Ditkoff's book, "Why Don't Your Eyelashes Grow?: Curious Questions Kids Ask About the Human Body." Republished here with permission.