Dolphins do not breathe underwater. Before they dive, they take a big breath. They hold it while they are underwater.
Dolphins are mammals. Just like all other mammals, they require air to breathe. They breathe air through their blowhole. They return to the surface every few minutes to take another breath. While underwater, the blowhole closes so no water can get into their lungs and cause them to drown.
Since dolphins cannot breathe under water, they have adapted the ability to eat underwater. No air is taken in through the dolphin's mouth, so they are able to catch fish without drowning.