Brands
Brands are only obtained by mutation or breeding. To get a brand mutation, the pengi owner must have an expert badge related to the brand.
Expert badges are earned by pengis doing tasks at task stations.
Once the pengi owner has unlocked the expert badge, each time a pengi lays an egg it has a small chance to obtain that brand.
Brands can pass to offspring, but it is rare.
Brands were originally just meant to be a bit of fun and add an extra dimension to the breedable game, however I am exploring ways of making them more meaningful. For now, it is possible to disable brand mutation entirely via your web portal (and enable it again if you wish).
To see which task stations give which brands, see the task-station page. This page lists all the task stations, and which badges each task station gives (and therefore which brand).