Ore is a resouce which can be mined from Asteroids by using Mining Lasers, and Logistics Drones to collect the Ore. They currently serve three purposes: selling to traders, completion of Contracts, and smelting to Ingots.
Precious Ore
Precious Ore can be mined outside of Contracts at any time from Asteroids, and can be smelted into precious metal. It sells for 2,100 or 2,200 per stack.
Tillium Ore
Tillium Ore only appears during Contracts to extract it, however, it can continue to be mined beyond the delivery amount. It is noted in game that Tillium Ore is refined into Fuel, however it was states in update 2, its refined into FTL Fuel, presumably FTL charges.
Tilium ore can be refined via a smelter into Refined Tillium