The Leaf Stone can be crafted from 9 Leaf stone shards. Leaf stone shards can be found on trees in forests and jungles. They are leaf blocks with bright yellow patches on them. Each block will drop 1 shard so you need to find 9 of them. When you have the leaf stone, there are currently 3 uses for it.
First, it can evolve Gloom into Vileplume. Second, it can turn Weepinbell into Victreebel. Finally, it can evolve Exeggcute into Exeggutor. In the future it will also be able to evolve Nuzleaf into Shiftry and Pansage into Simisage. Send out the compatible Pokémon, then hold the leaf stone and right-click the Pokémon. It will evolve into the corresponding evolution.