Eevee Evolutions

Eevee is an "Evolution Pokémon". It is called this because it can evolve into one of several kinds of Pokémon by usage of various methods.
It can evolve three ways by evolutionary stone, twice through Happiness, and twice again by having Eevee gain a level in Eterna Forest and on Route 217, respectively.

The first three "Eeveeloutions" as they are sometimes called, are simply achieved by using one of three Stones on an Eevee.
Flareon, a Fire-type, is the result of an Eevee being exposed to a Fire Stone.
Vaporeon, a Water-type, is the result of an Eevee's exposure to a Water Stone.
Jolteon, an Electric-type, is the result of exposing an Eevee to a Thunder Stone.

The next two Eevee evolutions are achieved as the result of an Eevee trusting a Pokémon Trainer very much by raising their Happiness level to the maximum (represented by two big hearts on the Friendship Checker application on your Pokétch), then being leveled up during a certian time of the day depending on which one you would like.

Espeon, a Psychic-type, evolves during the day between the hours of 4am and 8pm.
Umbreon, a Dark-type, evolves during the night between the hours of 8pm and 4am.

Finally, the last two of the Eevee line evolve differently depending on the environment around them.

Leafeon, a Grass-type, evolves by leveling up an Eevee at the patch of grass near the moss-covered rock in the southwestern part of Eterna Forest.
Glaceon, an Ice-type, evolves by leveling up an Eevee at the patch of grass near the ice-covered rock on the northeastern part of Route 217.