High happiness (also referred to as friendship) at night, when not near an Ice Rock or Moss Rock. No stone is required.
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To make Eevee evolve into a Vaporeon, you have to expose it to a Water Stone.
To make Eevee evolve into a Flareon, you have to expose it to a Fire Stone.
use a fire stone
You can't make eevee evolve into espeon by leveling it up. You have to give it a certain stone. I think it might be a moonstone, but maybe that makes eevee evolve into umbreon.
A fire stone,water stone,ice stone,dusk stone,leaf stone,thunder stone,and dark stone will evolve him/her.