If you evolve an Eevee while it is your buddy pokemon and you have walked at least 10km with it, it will turn to Umbreon during the night and to Espeon during the day. But what are the exact timeframes here, i.e. what times are considered day and night by Pokemon Go? I haven't been able to find an exact answer to this.
This likely uses the same mechanism the game uses to determine whether to show "day" or "night" maps. I believe it gets the local sunset/sunrise time from the phone. In this case the evolution will be Espeon from sunrise to sunset, and Umbreon from sunset to sunrise.
Update: Now that the evolve button shows what you're going to get (if it's not a random draw among the first three), as the other answer here describes, that's a much better way to tell when to evolve.
If you are mostly concerned with the time you should evolve: When you go to the Eevee you want to evolve, located on the “evolve button” a picture (or silhouette if you haven’t registered that Pokémon yet) will show up if you have walked 10k with that Eevee. Indicating what Pokémon it will evolve into. It will either be in the shape of Espeon (two-pronged tail) or Umbreon. This is a simple way to know when to evolve.