A Curse spell targets the player it will enchant like any other Aura spell, and a Curse stays on the battlefield like any other Aura. If the enchanted player gains protection from the Curse’s color (or any other characteristic the Curse has), the Curse will be put into its owner’s graveyard.
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The Curse subtype has no inherent rules meaning, but two cards in the Innistrad set (Bitterheart Witch and Witchbane Orb) refer to Curses.