Pokémon X and Y Walkthrough, Pokémon Move Sets: Espeon
(Please note that the recommendations below are largely made for in-game play. An Espeon in a competitive environment will likely make use of different moves to accommodate for smarter, more adaptive opponents. If you've got different movesets in mind for an Espeon, share 'em in the comments.)
The light half of the second generationf of Eeveevolutions, Espeon was born to be a sweeper. There are no other Eeveevolutions so adept at wiping out entire times like Espeon, and though there are better sweepers on the market, they never seem to have the same level of panache.
Espeon, like the other Eeveevolutions, starts off as an Eevee. To turn it into an Espeon you need to raise its happiness to maximum, likely through extensive use of Pokemon-Amie. Once that's done, level Eevee during the day - not the night, the day! - and it will turn into an Espeon. Be careful, as doing otherwise will get you an Umbreon instead.
Bug, Dark, Ghost.
As mentioned above, Espeon is a born sweeper. It has a Special Attack that eclipses the base stats of many other pokémon, and this Special Attack is supported by a considerable Speed stat. Special Defense comes in a decent, though kinda distant, third. Everything else in Espeon's repertoire comes last, reflecting Eevee's nature as the opposite of Umbreon: offensively powerful, but quite frail.
Synchronize: If the pokémon is poisoned, burned or paralyzed, the opponent who inflicted the status ailment will similarly be poisoned, burned or paralyzed. Though not quite as useful on Espeon since you're usually going for OHKOs, Synchronize has its uses against speedy Thunder Wave users.
Magic Bounce: Non-damaging moves that would inflict status ailments, stat changes or field effects are reflected back at the user. This is useful for a myriad of reasons, though it's probably best used for essentially making your team immune to Spikes, Toxic Spikes, and Stealth Rock while it's on the field. Note that Magic Bounce only reflects moves as they're used; they won't get rid of Spikes and the like if they've already been deployed. Highly recommended. Magic Bounce is a hidden ability.
Espeon is a sweeper and it should be equipped as such. Harkening back to the old days, Psychic is your primary weapon for Espeon. This move alone can usually wipe out opponents whether it's super effective or not. You can then outfit Espeon with three other non-STAB moves to fill in holes in your team. Shadow Ball is popular for wiping out ghost-types that would otherwise be a threat; Grass Knot wipes out a horde of heavier rock-, ground-, and water-types; Dazzling Gleam is a new weapon to deploy against dragon-types, among others.
Since Espeon is such a popular sweeper, you may also want to consider using it to Baton Pass. Keep Psychic as a STAB move in case you have the opportunity to inflict super effective damage; otherwise, give your Espeon some boosting moves. Reflect is your best bet for warding off the physical hits that would otherwise knock it out in one turn, then use Calm Mind a few times until Espeon is maxed out. Baton Pass the results to a different sweeper and watch the fireworks. You can use this build on a Synchronize Espeon, but it's better reserved for one with Magic Bounce.
EV / Super Training
Most Espeons will want to boost their two most impressive stats, Special Attack and Speed, as high as possible. The harder and faster you can hit with your moves, particularly moves that do not benefit from STAB, the better. An Espeon that goes the Baton Pass route, on the other hand, is better outfitted with HP / Defense and Speed EVs. Max out either HP or Defense, then split the remainder between Speed and the stat you didn't cover.
Catching an Espeon
Those blasted Eeveevolutions never show up in the wild, unless you get really lucky in someone's Friend Safari. To get your Espeon you'll have to track down, and evolve, an Eevee. Have a look on Route 10, preferably in the yellow flowers, and you'll find one eventually.