So I found it quite hard to grasp the Immunity Island rules so after some help from the community, I would like to do this post for those who are still confused. So, here we go (please flag me for any mistakes).
Act 1: A member of the losing tribe is sent to Immunity Island. Once on Immunity Island, the player receives an Immunity Necklace and the mechanics for that round's mini-game.
Scenario 1: The castaway chooses to play the minigame. Regardless of the result, this player will keep his immunity, but cannot vote.
Scenario 2: The castaway chooses to leave. They will immediately return to camp to be able to strategize and vote at Tribal Council, but the player must assign the Immunity Necklace to someone else.
Act 2: The player stays on Immunity Island to play the minigame. They will stay on Immunity Island until Tribal Council, not being able to take part in the strategizing at camp.
Scenario 1: The player wins the minigame, obtaining themselves an advantage in the game.
Scenario 2: The player loses the minigame, receiving a pre-determined punishment.
Scenario 3: For choosing to stay on Immunity Island, the player may also look for a clue to their tribe's Hidden Immunity Idol hidden back at camp.*** It remains to be seen if refusing to stay on Immunity Island also bars the player from looking for the clue to their tribe's idol back at camp.
Hope it clears everything up.