Ore is an element obtained on occassion when you defeat the boss of a stage. This ore can be refined into a variety of different styles and the contents vary. The Ore is a way to get Gears in order to power up Pokémon to the best they can be
Each stage has got its own Pokémon focus and, as each Pokémon has a type, it gets an Ore focus as well. This means that levels focusing on a Water-type Pokémon would have a Water-type ore focus, allowing for farming of certain levels in an attempt to get the gears you desire. These also have an increase in yields as you rank up the island
The three different ore have a different length of time to refine and you can only refine one ore at a time. This can however be added to if you buy the Refining Pack on the Shop, giving you a second slot to refine ore in. You can also pay Poké Diamonds to completely refine the ore or use a ticket to remove an hour from the counter.
If you have all three ore slots filled and get more ore, you get given a choice of dismissing the new ore, using Poké Diamonds to clear a space, or dismissing an ore you have in one of the slots that isn't getting refined. This final option can only be done 10 times a day.