Instructions
Go into the Celadon City Pokemon Center and deposit one of the Pokemon from your party into "Bill's PC" in order to make room for Eevee. The PC is located in the far right corner of the Pokemon Center.
Climb to the top floor of the Celadon City Department condominiums and speak with the gentleman in the room. You must enter the condominiums from the back entrance in order to reach the top floor. Once you finish speaking with the gentleman, he will gift you an Eevee, which will end up in your party.
Purchase Carbos and Calcium items from the second floor of the Celadon City Department Store's "vitamins and minerals" section. Applying Carbos and Calcium to Eevee during battle or while walking will raise the happiness level of the Pokemon and cause it to evolve into Espeon faster.
Continue to walk with Eevee in your party during the day. If your Eevee accidentally evolves at night, it will turn into an Umbreon. If this happens, press and hold the "B" button to cancel the evolution process. Once your Eevee reaches a certain happiness level, it will evolve into Espeon.