I caught an Eevee, which of its evolutions will it evolve into?

According to this post on the /r/pokemongo subreddit, you can force your Eevee to evolve a certain way by changing its nickname before evolving it to:

• Pyro to evolve into Flareon.
• Sparky to evolve into Jolteon.
• Rainer to evolve into Vaporeon.

(These are the names of Eevee Brothers from the TV series/Manga)

You can also, as of the Generation II update, use nicknames to evolve into a certain Eeveelution, but like before, this works only once.

• Tamao to evolve into Umbreon.
• Sakura to evolve into Espeon.

This now works with the Generation IV evolutions Leafeon and Glaceon. When renaming Eevee the first time, the evolve button with display the evolution it will evolve into rather than a question mark.

• Linnea to evolve into Leafeon
• Rea to evolve into Glaceon

This technique will only work once per evolution. For example, if you have already evolved an Eevee named Rainer into a Vaporeon, evolving another Eevee named Rainer will result in a random evolution.

Evolving without the naming tricks will give you Flareon, Jolteon or Vaporeon by random chance.

Also, according to another reddit thread, by walking 10km with Eevee as a buddy you will get Espeon when evolving during the day and Umbreon when evolving during the night.

Vaporeon, Jolteon, Flareon

Evolving an Eevee without meeting any other requirements/prerequisites, will randomly result in either a Vaporeon, Jolteon, or Flareon with a 1/3 chance of either of them.

Espeon, Umbreon

If you keep Eevee as your buddy for 10km, earn at least 2 candy with it whilst it's your buddy, and then evolve it whilst it's your buddy, it will evolve into either:

• Espeon if evolved during the day.
• Umbreon if evolved during the night.

The 10km and 2 candy requirement is counted across all buddy uses of Eevee. You can earn the kilometers and candy, either intermittently or all at once, whilst changing buddies or keeping it the same.

As long as a total of 10km and 2 candy has been earned whilst Eevee has been a buddy, and you evolve it whilst the Eevee is currently set to your buddy. It will override the ability to evolve into Vaporeon/Jolteon/Flareon whilst it's your buddy, but not Leafeon/Glaceon if near the relevant lure.

Day vs night is determined by the game screen in the overworld/map. It's daytime when it is bright, and nighttime when it's dark:

The times when the game screen changes over varies depending on your GPS and time location.

Leafeon, Glaceon

If you purchase a Moss or Icy Rock from the shop (for 200 coins), place it on a Pokestop (like a Lure), spin the Pokestop, and then evolve Eevee whilst in the radius of the Rock, it will evolve into Leafeon (Moss Rock) or Glaceon (Icy Rock).

The silhouette of Leafeon/Glaceon will appear if these requirements are met prior to evolution - if it doesn't, the requirements are not met and it would evolve into Vaporeon/Jolteon/Flareon (or Espeon/Umbreon if their conditions are met).

Syvleon

Earn at least 70 hearts with your Eevee so that it becomes a Great Buddy (also works with Ultra and Best buddy). Afterwards, whilst Eevee your buddy, you are given the option to evolve it into Sylveon. Unlike the other evolution methods, the option to evolve into Sylveon is available simultaneously alongisde Espeon/Umbreon and Leafeon/Glaceon if those conditions are met.

You can also choose a nickname to force a specific evolution, but this only works once per nickname/evolution.

In addition to the first three evolutions mentioned by @Brant, with the addition of the second generation there are two more Eevee names that will trigger a guaranteed evolution once:

• Tamao for an Umbreon.
• Sakura for an Espeon.

According to this site, stones are available after the gen 2 update. Other than the naming tricks, there are other ways to control some evolution forms:

Walk with the Eevee for 10 kilometers and stay it as your buddy, then evolve it during: day – Espeon; night: Umbreon.

There may be other better ways to control evolutions, but haven't been found out by all trainers.