Archipelagos are a special time-limited type of adventures that are available for you in the game. More Archipelagos will unlock as you will progress through the game. At this moment there are four Archipelagos available in the game.
In order to travel to an Archipelago you need to press the map icon in the bottom-left corner of the screen:
After that, you can click on the active Archipelago and the archipelago map will open allowing you to chose an island to visit. Alternatively, you can press on the quest related to the Archipelago and then press "Go" button there.
In order to leave the Archipelago and return to your main island you need to press the door icon in the bottom-left part of the screen: