I'm updating this as the current is outdated, because new Eeveeloutions have been added.
NOTE: You still have to give Eevee 25 Eevee candies to evolve Eevee, no matter what method you do.
If you just give it 25 candies, it will evolve into either Vaporeon, Jolteon, or Flareon. You can name them to get a guaranteed evolution, or Mossy Lure Module guarantees that Eevee will evolve into Leafeon, and Glacial Lure Module guarantees that Eevee will evolve into Glaceon. Glacial Lure Modules and Mossy Lure Modules both cost 200 Pokecoins. If you walk 10 km with Eevee as your buddy, it will be guaranteed to evolve into Espeon, and if you evolve Eevee at night, it will evolve into Umbreon. Earning 70 hearts with Eevee as your buddy will guarantee that it evolves into Sylveon.
This is a list of all the names:
- Vaporeon: Rainer
- Jolteon: Sparky
- Flareon: Pyro
- Leafeon: Linnea
- Glaceon: Rea
- Espeon: Sakura
- Umbreon: Tamao
- Sylveon: Kira
Each name only works once per account.
Source