Nothing is done on the turn that Future Sight is executed. During the 3rd end-of-round phase subsequent to the use of Future Sight (the round of use plus the next two rounds), damage from Future Sight will be dealt. Any other attacks may be used in the meantime. If Future Sight is used during this time by the pokemon that used it before, or one that has switched in, it will automatically fail.
As implied in the preceding paragraph, the pokemon that used Future Sight doesn't need to still be in the fight when Future Sight deals damage. If another pokemon has taken the place of the using pokemon, both the Accuracy and Special Attack stats of the pokemon that actually used the move will be used for the purposes of determining the accuracy and damage (respectively, of course) of the attack.
The damage from Future Sight is effectively typeless. It will not be increased as per STAB if used by a Psychic pokemon and the damage will not be decreased against Dark or Steel-type pokemon.
Future Sight is affected by Brightpowder when the opposing pokémon is equipped with it.