Penguin feathers can be obtained by shearing a tame penguin. Each penguin drops 0-2 feathers when sheared, according to its color:

The penguin feathers can be crafted into many items, listed here:

When a penguin is sheared, there's a 33% chance that it will become angry and attack you, in which case you can pacify it by feeding it more raw fish. This chance is reduced to 20% with Penguin Shears. An untamed penguin will always become mad if you attempt to shear it.

Flame Penguins can't be tamed but they can be sheared as long as they're not angry. Since a Flame Penguin will become angry if it sees an undisguised player, you have to be wearing a full suit of any penguin armor to shear it. Then if it becomes angry at being sheared, you can pacify it by feeding it Magma Fish (as long as you are still wearing your penguin armor). Flame Penguins will always become angry on shearing unless you are using Penguin Shears, in which case it will only happen half the time. Instead of feathers, Flame Penguins drop Flame Penguin Scales.

Cloud Penguins are also untameable, but they can still be sheared if they are not trying to fly away from you. This can be achieved by disguising yourself with a full suit of penguin armor consisting of any combination of Flame and Cloud pieces. Upon being sheared, a Cloud Penguin will fly in the oppositie direction for a considerable distance and will be unshearable for 30 seconds. This amount of time is cut in half if Penguin Shears are used. Instead of feathers, Cloud Penguins drop Cloud Penguin Down.

Ninja Penguins can't be tamed or sheared.