Instructions
Obtain an Eevee. Eevee's location varies by game, but typically you are only given one. Trade for a second Eevee, or breed the first with Ditto to get a second one.
Raise Eevee's happiness to the maximum. Use Eevee in battles, win frequently and don't let it faint. Feed Eevee healing drinks like Lemonade, MooMoo Milk or Soda Pop. Some of the newer games have a feature where you can take Eevee for a massage or haircut daily, which also raise its happiness. Leveling up, feeding it particular berries -- depending on its nature -- and giving it vitamins like Carbos or Protein make it happier as well.
Check Eevee's happiness. In "Pokemon Diamond," "Pearl" and "Platinum," you can check happiness with a Poketch app received in the Eterna City Pokemon Center. In "HeartGold" and "SoulSilver," a woman in Goldenrod City will tell you how happy your Eevee feels.
Level Eevee up during the daytime, and as long as its happiness is over 220 it will evolve into Espeon. You can level up Eevee by battling or using a rare candy.
Level your Eevee up at night for it to evolve into Umbreon. Again, level doesn't matter -- but happiness must be at least 220.