Emeralds are the rarest mineral in Minecraft. They only appear in the bottom 8 layers of the map and in branches of 1-5. They require an Iron Pickaxe or better to mine.
Emerald tools have twice the durability of Diamond tools (3124 uses) and will do anything twice as fast. They are the absolute hardest mineral to find, but they are worth the search. You can trade Emeralds to Villagers to get rare items.