The Pokémon games have eight different Eevee evolutions.
These evolutions are commonly called the Eeveelutions. The original three are Vaporeon, Jolteon, and Flareon, which evolve using Water, Thunder, and Fire Stones. Espeon and Umbreon were added in Generation II, followed by Leafeon and Glaceon in Generation IV, and Sylveon in Generation VI.
Eevee is unusual because its evolutionary possibilities span multiple types and use several different conditions. Some require an evolutionary stone, while others depend on friendship, time of day, a specific location, or knowledge of a Fairy-type move.
The total is eight, not nine. Eevee itself is not counted as an evolution, and forms or special battle transformations do not create additional Eeveelutions. The number can also be confused with the total number of Pokémon species introduced in a generation, which is unrelated.