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Pokemon Island is basically a follow-up to Pokemon Snap where the island that Todd visiting ended up becoming a popular resort for trainers years later.
The best way to describe PI's gameflow is that it's Postgame: The Game. Instead of fighting Gym Leaders + Evil Team and E4, you catch Pokemon at many different areas of the Island so you can use them to challenge the Single and Double Colosseums and raise your trainer rank. As you raise your rank, more island areas and special facilities (among other things, like rivals and key items that replace HMs) in the main city become available for use.