Dust is a resource obtained by burning weapons to get their respective dust counterparts.
Dust comes in three forms, Lesser Dust, Greater Dust, and Powerful Dust which can be obtained by burning 1-3 star weapons, 4 star weapons, and 5 star weapons respectively.
The amount of dust you receive depends on the rarity of the weapon burned, and any on-going promotions such as a reforge event.
Under normal circumstances, players are expected to receive the following amounts of dust when burning a weapon:
Burnt Weapon Rarity
Dust can also be given out as a reward in Raids, and potentially other sources as more features are developed.
Dust functions the same as as burning a similar rarity weapon for reforging, with the added benefit of batched transactions rather than manually burning one weapon at a time. There is also a side benefit of cleaning up a player's blacksmith page instead of showing hundreds of weapons.
More information can be found in the Reforging page here: