If Ixalan's Binding leaves the battlefield before its triggered ability resolves, the target permanent won't be exiled.

Auras attached to the exiled permanent will be put into their owners' graveyards. Any Equipment will become unattached and remain on the battlefield. Any counters on the exiled permanent will cease to exist.

If a token is exiled this way, it will cease to exist and won't return to the battlefield.

The exiled card returns to the battlefield immediately after Ixalan's Binding leaves the battlefield. Nothing happens between the two events, including state-based actions.

If the exiled card has a different name in exile than it did on the battlefield, Ixalan's Binding stops players from casting spells with the name of that card as it exists in exile.

If the exiled card is a land card that's become a nonland permanent, Ixalan's Binding won't stop players from playing lands with that name.

If there is no exiled card (perhaps because the exiled permanent was a token or was a commander that moved to the command zone in the Commander variant), Ixalan's Binding won't stop players from casting spells.

In a multiplayer game, if Ixalan's Binding's owner leaves the game and another player owns the exiled card, the exiled card will return to the battlefield under its owner's control. Because the one-shot effect that returns the card isn't an ability that goes on the stack, it won't cease to exist along with the leaving player's spells and abilities on the stack.

There are many important moments in the story, but the most crucial—called "story spotlights"—are shown on cards. You can read more about these events in the official Magic fiction at http://www.mtgstory.com.