A weapon that has been destroyed through Enhancement or Enchantment can usually be restored with AP by visiting the Strange Traveler and clicking "Other", then "Destroyed Item".
Restored items can be reforged by visiting Ferghus. This allows the item to be enhanced again, as long as the restored item is +10 - +14. However, it will still be destroyed permanently if it fails the next Enchantment.
If a weapon has been "Permanently Destroyed" this means that the weapon is unable to be revived by any means including with AP. If a weapon has been restored before, it will be Permanently Destroyed when the next Enchantment is failed.
In short, after the newest update, Enhancements can be failed as many times as you'd like as long as you reforge in between.
Enchantments will permanently destroy an item after it has already been restored, regardless of reforging.
You will know if a weapon or armor has been restored by noting the following tag in the tooltip:
Please keep in mind that the Strange Traveler only stores up to 10 destroyed items. If more than 10 are destroyed, the item that was destroyed first will be removed from the list.
Please know that this feature can only be accessed by characters that have reached level 90. If you have destroyed your weapon, but you are under level 90, you will have to wait until your character reaches level 90.
Note: Once restored, it will be forever bound to the character.