I feel qualified to answer this question because I've played XY extensively, in normal playthroughs and various Nuzlockes.
The Eeveelutions available in X and Y are Vaporeon, Jolteon, Flareon, Espeon, Umbreon, Leafeon, Glaceon, and Sylveon. Umbreon is too defensive for in-game, where offense matters much more. The Mossy and Ice Rock come too late in the game (Route 20 and the Frost Cavern) for Leafeon and Glaceon to be of much use, especially considering that excellent Pokemon of both type like Gogoat and Lapras are available earlier. Flareon has far too shallow a movepool to put in any work. This leaves the viable contenders at Vaporeon, Jolteon, Espeon, and Sylveon. Let's evaluate the four.
Vaporeon
Pros
- Has the most HP of the four
- Ties for second-highest Special Defense
- Effective against Grant, Wulfric, and Malva
- Learns moves like Surf, Strength, Waterfall, Shadow Ball, Ice Beam, and Scald
- Great HM holder
- Water Absorb gives it a nice immunity
Cons
- Water Stone typically comes too late for Grant, unless you use Super Training
- Second-worst in terms of Speed
- 65 base Defense
- Water types are abundant -- Lapras and Tentacruel are among those arguably better than Vaporeon
- Good matchups aren't too common
Jolteon
Pros
- Highest Speed
- Second-highest Special Attack
- Volt Absorb gives it a nice immunity
- Earlier evolution than Vaporeon and (probably) Espeon
- Learns moves like Thunderbolt, Strength, Shadow Ball, Volt Switch, and Thunder Wave
- Effective against Viola, Siebold, and bits of other important Trainer teams
Cons
- Very frail
- Most of its attacks come off its weaker physical Attack
- Shallow special movepool
- Isn't inherently good against any type specialists -- must pick and choose who it fights
- Comes too late for Viola
Espeon
Pros
- Highest Special Attack of the four
- Second-highest Speed
- Synchronize is decent
- Learns moves like Psychic, Morning Sun, Shadow Ball, Grass Knot, Calm Mind, and Dazzling Gleam
- Effective against Korrina and, with Dazzling Gleam + Shadow Ball, Olympia and Drasna
Cons
- Physically frail
- Happiness evolutions are tedious
- Learns only the most useless HM, Cut
- Synchronize isn't as good as immunity Abilities
Sylveon
Pros
- Highest Special Defense
- Second-highest HP
- Solid Special Attack
- Potentially the first Eeveelution obtainable
- Learns moves like Moonblast, Light Screen, Reflect, Shadow Ball and Calm Mind
- Effective against Grant, Korrina, Drasna, and Diantha
Cons
- Steel and Poison types are very common
- Shallow special movepool
- Even slower than Vaporeon
- Pokemon-Amie may give unwanted Affection boosts (and waste your time)
- Can't take physical hits too well
Ready for the cliche answer? The best Eeveelution depends on the rest of your team. If you need Electric coverage, use a Jolteon. If you have a Lapras that works well, there's no need to raise a Vaporeon.
If we judge objectively by availability, positive matchups, movepool, and stats (without having other team members already chosen), then the best Eeveelution is probably Espeon. While the Happiness method of evolution may be a pain, its incredibly high Special Attack, Speed, varied movepool, and positive major matchups outweight the downsides. Obviously in-game any Pokemon can work, and you're free to use whatever Eeveelution you want.
Note: If you want the best physical attacking Eeveelution (as the four aforementioned ones all lean special), then choose Leafeon. It comes late, but the only other choice is Flareon, which is so bad it's painful.
A good set for Espeon would look something along the lines of:
Espeon @ Twisted Spoon
Ability: Synchronize
EVs: 252 SpA / 252 Spe / 4 HP
Modest Nature
- Psychic
- Dazzling Gleam
- Shadow Ball
- Grass Knot / Morning Sun / Calm Mind
