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.

Each of the Curses can be attached to any player, including the player who cast the Curse.

Curse is an enchantment type, not a creature type (or any other kind of subtype).

The ability won't trigger when a creature attacks a planeswalker controlled by the enchanted player.

The Curse subtype has no inherent rules meaning, but two cards in the Innistrad set (Bitterheart Witch and Witchbane Orb) refer to Curses.