Summarize:

After having a full friendship meter and leveling during the day, Eevee will evolve into Espeon.  You can achieve this by battling with your Eevee on the team (without it fainting), keeping your Eevee on the team, giving it berries and protein, as well as giving it haircuts and taking it to the National Park. You can get a general idea of your Eevee's happiness by talking to the woman in Goldenrod City. You can find her north of the Bike Shop and on the east side of town. If she says, "It looks really happy! It must love you a lot," your Eevee is ready to level and evolve. Only interact with your Eevee between 4:00 AM and 8:00 PM, as any time outside of that means your Eevee will evolve into an Umbreon. After your Eevee has a high friendship level, you will want to battle with your Eevee for it to level at least once. After leveling in the daytime (4:00 AM - 8:00 PM), Eevee will evolve into Espeon.
Get a high friendship with your Eevee. Level your Eevee between 4:00 AM and 8:00 PM.