What happens to spells cast from Nightveil Specter's ability?
Casting a spell exiled with Nightveil Specter follows almost all steps mentioned in section '6. Spells, Abilities, and Effects' of the comprehensive rules, including


  608.2k As the final part of an instant or sorcery spell's resolution, the spell is put into its owner's graveyard. As the final part of an ability's resolution, the ability is removed from the stack and ceases to exist.


So, yes, it's put into its owner's graveyard, unless the card itself specifies otherwise (e.g. Time Reversal or Beacon of Immortality).