Pokémon X and Y Walkthrough, Pokémon Move Sets: Umbreon
(Please note that the recommendations below are largely made for in-game play. An Umbreon in a competitive environment will likely make use of different moves to accommodate for smarter, more adaptive opponents. If you've got a different moveset for your Umbreon that you'd like to share, shove it down in the comments.)
One of two second generation Eeveevolutions and arguably one of the most popular overall, Umbreon is a fan favourite for a reason. Eschewing the normal dark-type style of overwhelming offensive power, Umbreon is a tricksy, bulky creature that is better at irritating opponents than straight-out taking them down. This is a pokémon to pick if you love to pester your opponents to death.
As an Eeveevolution, Umbreon does not have a standard leveling trick to its evolution. To evolve an Eevee into an Umbreon you must first max out its friendliness with you, usually through a combination of Pokemon-Amie, grooming, putting it in the lead during fights, and making sure it doesn't get knocked out. (For EV purposes, I recommend Pokemon-Amie pampering.) Once your Eevee is nice and happy, level it during the night to get an Umbreon. Be very careful on this last point - if you level it during the day by accident you'll get an Espeon instead.
Dark, Ghost. Immune to Psychic.
Bug, Fairy, Fighting.
Umbreon is overwhelmingly a defensive pokémon. Its Special Defense is extremely beefy; its Defense is similar but not quite as high; its HP is daunting. Everything else - Attack, Special Attack, Speed - is below average. As such, Umbreon is very hard to kill, but it also has a hard time killing other pokémon... or at least it does if you try to load it down with normal attacks.
Synchronize: If the pokémon receives a status ailment - poisoning, burning, or freezing - its opponent will be inflicted that same ailment, assuming it's not protected by an ability or its very nature. This is an extremely handy ability, as using Toxic is one of the most reliable ways for bringing down tanks like Umbreon.
Inner Focus: The pokémon is immune to flinching. Always fun, but Synchronize is so, sooooo much better. This is a hidden ability.
Umbreon is practically built to poison other pokémon. Toxic will be its primary weapon, allowing you to slowly chip away at even the bulkiest of opponents with relative impunity. Mean Look prevents switching, Confuse Ray is debilitating and has the potential for doing extra damage, Curse ups your Defense against fighting-types, and Moonlight will keep your Umbreon nice and healthy. You can also swap in Protect or Substitute as a way of offsetting damage above and beyond Moonlight. Including at least one STAB move to replace Confuse Ray isn't a terrible idea either, in which case I'd recommend Payback. It'll nicely complement Umbreon's poor Speed stat on those occasions when Umbreon goes second in a round, which, properly EV trained, it usually will.
EV / Super Training
Defense, defense, defense. Umbreon's already got some hefty fortifications in place, and they'll become ridiculously potent if you add EVs to its Defense and Special Defense. Alternatively, you can pile the points into HP alone and make the most of Payback by chucking the rest of your points into Attack, though in this case I'd recommend dividing the remainder between the two defensive stats instead. Payback is fun, but survivability should be your Umbreon's number one priority in most matches.
Catching an Umbreon
Umbreon, like the other Eeveevolutions, does not appear in the wild. To get one you'll have to track down and evolve an Eevee. You can find these elusive creatures on Route 10, near the menhir fields north of Cyllage City. Check the yellow flower plots for a better chance at coming across an Eevee.