There are different types of islands in the game. The more you play, the more islands you can visit.
First, there's your Home Island. You can build, plant crops and complete the requests from the shaman and the merchant there. Home Island is available in the game from the very beginning but you can also explore it and open different areas there.
Then there are quest islands. They become available as you progress in the game. Sometimes you need to reach a certain level to open them and sometimes you need to complete a certain important quest on your Home Island. Here you can find more information about the opening of new islands.
And last but not least, there are event islands. These are unique islands that appear in the game only for a certain amount of time. Each one of them is different and has its own interesting story.
You can also compete with other players in the leaderboard by completing the tasks there and after an event island is gone, you will receive a special reward. You can collect a special energy item there that can be brought to your Home Island.
Please note that event islands are only available for a certain amount of time. You can always see the remaining time on the icon with the island's tasks. When time is up, the island closes and all the resources and treasures left there disappear forever.