Dolphins are mammals because they have all of the major characteristics of mammals; they breathe through lungs, they are warm-blooded, they produce milk for their offspring and they have hair. There is also evolutionary evidence that connects dolphins and whales to other mammals.
Although dolphins and whales don't have easily visible hair or fur, they have hair before they are born. In addition, a few adults have whiskers or hairs on the top of their heads.
Another sign that dolphins are mammals is the fact that they have blowholes instead of gills. With gills, fish are able to breathe underwater, but with a blowhole, dolphins must return to the surface to breathe air.
Dolphins also are unlike fish in that they can regulate their own body temperature. This is because they are warm-blooded, while fish are cold-blooded and change temperature with the water around them.
Dolphins also have tiny limb-like appendages attached to the backs of their skeletons, which are hidden within their bodies near the tail. These "limbs" suggest that dolphins are descendants of mammals that walked into the sea and eventually evolved to become dolphins. In fact, the closest living relative of dolphins, other than whale, is the hippopotamus.