Yes, penguins do have nests. All the 17 species of penguins have nests; some on bare ice or some build nest using stones or stay in guanos. Species like Emperor and King Penguin don’t build any nest as such; they breed and lay eggs on open land.
The type of nests that the penguins make depends on the surrounding and the type of region they live. As any other birds, nest building is the crucial and significant part of the penguin’s action. Penguins use their nest mainly for breeding.