As an additional cost to cast a spell with gift, you can promise the listed gift to an opponent. That opponent is chosen as part of that additional cost. The gift isn't given at this time; rather, it's given at a later time based on whether or not the spell is a permanent spell.

If a spell for which the gift was promised is countered, doesn't resolve (perhaps because all of its targets are illegal), or is otherwise removed from the stack, the gift won't be given. None of its other effects will happen either.

"Gift a Treasure" causes the chosen opponent to create a Treasure token.

If you copy a spell for which the gift was promised, the gift was also promised to the same player for the copy. If a card or token enters as a copy of a permanent that's already on the battlefield, the gift isn't promised for that new permanent, even if it was promised for the original.

For instants and sorceries with gift, the gift is given to the appropriate opponent as part of the resolution of the spell. This happens before any of the spell's other effects would take place.

You can't pay a gift cost more than once.