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.
1So when the game uses the nighttime/blue ambience it's considered night? Apr 14, 2017 at 19:38
I haven't confirmed this, but it's the simplest hypothesis.– pjmorseApr 16, 2017 at 14:39
Well. I have read news once that it has been confirmed. I think it was on SilphRoad or similar– XetrovApr 21, 2017 at 16:39
To add onto this, night and day will be different depending where you live, time of year, etc. Very easy to distinguish in the game though. Apr 24, 2017 at 4:17
It definitely doesn't use local sunset time, though maybe it picks some number related to it. It's always gotten "dark" between 7 and 8 for me, but in reality, it's dark at 4 in the winter and 10 in the summer.– KatJul 19, 2017 at 0:33
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.
I think this was added in 2019 when Leafeon and Glaceon became available, and it is a way better way to tell what you're going to get from an Eevee evolution (or if it's going to pick one of the first three at random).– pjmorseJun 17, 2020 at 15:22