All animals need oxygen to survive, but they don't all get it in the same way. Fish, for example, breathe through gills, but seals don't have gills. As mammals, they have lungs that require breathing air to replenish the oxygen. While they don't breathe underwater, seals have several adaptations that help them function for long periods without breathing.
Regardless of whether an animal lives on land, in the water or in a combination of the two, one of the main defining characteristics of mammals is that they use lungs to breathe air and absorb oxygen. Mammals also give live birth, have hair and are warm-blooded. Other animals, such as frogs, have lungs, but they don't meet the other requirements to be classified as mammals -- they also diffuse oxygen through their skin. Seals, however, meet all the requirements, which means they only get oxygen from breathing air.
Diffusion vs. Breathing
Several animals use diffusion, at least in part, to bring in oxygen. Fish, for example, have gills that filter oxygen out of the water, transferring it to tiny capillaries. From there, the capillaries transmit oxygen to larger blood vessels and throughout the body. Breathing with lungs, like seals do, is different. Seals have tubes that lead from their mouth and nose to their lungs. They suck air into their lungs, where the alveoli help with gas exchange. The circulatory system transmits oxygen throughout the body. They use those same tubes to exhale carbon dioxide waste; in animals that use diffusion, the capillaries release it back into the water or the surrounding air without going through the lungs.
All mammals breathe in basically the same way, but some store oxygen more efficiently. Seals store oxygen in their blood and muscles, not necessarily in their lungs. They breathe out before they dive -- not holding their breath like humans do -- and rely on stored oxygen reserves to sustain them underwater. This oxygen storage rate varies between seal species. Some tend to dive shallow and stay underwater for 10 minutes or less, while others go deeper, staying under for as long as an hour without breathing.
Seals use some special adaptations to help them survive without breathing underwater. Their nostrils pinch completely closed, keeping water from entering the lungs. When they open their mouth to catch prey underwater, their tongue moves back to cover the throat so no water can enter. They have efficient lung systems as well; each breath replaces about 90 percent of the body's oxygen. In comparison, humans replace about 20 percent with each breath.