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Deals damage at the end of the second turn after it is used (if we say the move is used on the first turn, then the attack hits on the third turn). Since it hits at the end of the turn, it is not blocked by Protect or Detect. It does not bypass Focus Sash or Sturdy. Contrary to previous generations, it no longer deals typeless damage, so it is affected by STAB but also by weakness and resistance of the Pokemon it hits. Its power can be boosted by Gems, which activate the turn Doom Desire does damage. Life Orb now boosts Doom Desire's damage, and the user will suffer recoil damage. Since it's a Steel-type move, it no longer hits through Wonder Guard.
The damage dealt is calculated based on the Special Defense of the hit Pokemon and no longer from the one that was originally targeted. The damage is based on the Special Attack and Special Defense at the time of striking, not at the time it was used on the original target. This means that if you use Doom Desire on a Pokemon with 100 Special Defense, and then your opponent switches to a Pokemon with 300 Special Defense, the damage calculations will be done as though the target has 300 Special Defense. If you switch the user of Doom Desire out when the attack hits, then the user's Special Attack is calculated without boosts from items or abilities. Fails if this move or Future Sight is already in effect for the target's position, and multiple Doom Desires cannot be stacked.
The signature move of Jirachi, and for that reason only two Pokemon are able to use it. That said, none of them would have any reason to do so, even though it's the most powerful Steel-type move in the game. The mechanics just make it an undesirable option. Hitting three turns later, while at the same time giving the opponent enough room to simply switch to a resistant threat or a Pokemon with higher Special Defense, which thanks to the new mechanics is now calculated at the time it strikes, makes Doom Desire less appealing. Finally, the Steel-type isn't very prominent, offensively speaking, and the existence of Flash Cannon makes Doom Desire a useless option from a competitive point of view.
Flash Cannon is a less powerful Steel-type alternative that does not wait turns to do damage.
Future Sight is a much less powerful Psychic-type equivalent.