It can be obtained by mining in Black Rock Springs (Heartlands), Obelisks (Evershade Forest: Moonlight)
Mining obtains 1-3 ore but the XP goes from 70-280 (1 ore), 350-490 (2 ore), to 560 (3 ore).
Alicium Ore can sometimes be found from a Dragon.
...A defeated Lantern Monster may also leave one of these behind, but it must be a truly happy day for this to occur.
As an Ingredient
Those items use Alicium Ore in their recipe:
Alicium Ore is not craftable