Stamps are equivalent to in-game achievements awarded to players for completing a specific task or challenge in-game. Earnable stamps can be viewed from a player's Stamp Book, which will include a description of each individual stamp by hovering over them. The description will provide simple details of what is required of you to achieve the stamp.
Stamps are categorized as either Events, Activities, or Games stamps. Events Stamps include the stamps one can earn from participating in parties/events and meeting mascots in-game. Activities stamps include the stamps one can earn from completing general tasks around the island, several of which require the participation of multiple penguins working collectively as a group. For example, the Floor Filler stamp requires you to "dance in the Night Club with 25 penguins." Then finally, you have your Games stamps, which include stamps one can earn from playing the different minigames.
Stamps have varying levels of difficulty that are considered either easy (green), medium (yellow), hard (blue), or extreme (red) stamps.
If you are able to achieve all stamps for a particular minigame, you will receive a double coin bonus as reward while playing that minigame going forward.