Feeding it enough candies without a bit of intervention will result in a random evolution, but according to THIS site, and THIS site, and THIS site, you can pretty much choose whatever Eevee-lution you want.
The simplest method is to change your Eevee's nickname, then level it up with enough candy for it to evolve.
Here are the required nicknames:
Rename Eevee as Linnea to evolve into leaf-type Leafeon
Rename Eevee as Rea to evolve into ice-type Glaceon
Rename Eevee as Sakura to evolve into psychic-type Espeon
Rename Eevee as Tamao to evolve into dark-type Umbreon
Rename Eevee as Rainer to evolve into water-type Vaporeon
Rename Eevee as Sparky to evolve into lightning-type Jolteon
Rename Eevee as Pyro to evolve into fire-type Flareon
This only works once per each Eevee-lution. (You can only get one Flareon, one Vaporeon, and so on using this method.)
For Umbreon and Espeon, you can also choose to level them up with a Night/Day element involved.
Select Eevee as your buddy, walk with them for 10 km, and earn two candies. Then, with Eevee still as your buddy, evolve it. If you evolve it during the day, you'll get an Espeon. During the night will yield an Umbreon.
[Edited to include Glaceon and Leafeon]