Penguins stay warm through blubber. Just like whales, penguins have a layer of fat under their skin called "blubber". Overtop of this they are covered with fluffy "down" feathers and overtop of those they have their outer feathers which overlap to seal in warmth. Penguins rub oil from a gland onto their feathers to help make them waterproof and windproof.
Even so, penguins often need help to stay warm. In photos and video, you'll often see groups of penguins huddled shoulder to shoulder with their wings tight against their body keeping each other warm. As many as 5,000 penguins will bunch together to warm each other up.