(Please note that the recommendations below are largely made for in-game play. A Sylveon in a competitive environment will likely make use of different moves to accommodate for smarter, more adaptive opponents. If you love your Sylveon to death and want to share how awesome it is, let us all know what moves you used in the comments.)
The Eeveevolutions have always been popular with Pokémon fans, all the way back to the first generation, and it's always interesting to see how these single-type critters will compare to one another. Sylveon, the newest of the pack, may well go down as one of the best, given its powerful combination of typing, stats, and moves. It may look pretty, but don't underestimate this powerful pokémon for one second.
Sylveon is part of the Eevee line of evolutions, and is one of the few that does not rely on a stone of any kind to evolve. In order to get a Sylveon you need two things: a fairy-type move, gained early via Baby-Doll Eyes, and a minimum of two hearts in Pokemon-Amie. The next time your Eevee levels with these conditions in place it will turn into a Sylveon.
Bug, Dark, Fighting. Immune to Dragon.
Sylveon can be a fantastic pest, and its stats are largely to blame. Though neither the fastest nor strongest pokémon in the world, Sylveon has an outstanding amount of HP, a high Special Attack and an even higher Special Defense. It is very, very difficult to beat a Sylveon with special strikes, giving it a considerable advantage against most poison-type moves. The only downfall of Sylveon is its comparably low Defense score, which is half that of Special Defense.
Cute Charm: If the pokémon is hit with a physical move delivered by a pokémon of the opposite sex, it becomes infatuated. It will therefore only use moves against the target pokémon half of the time. Cute Charm can be utterly debilitating and force a switch if you get lucky... or it can be utterly useless. In short, don't depend on it.
Pixilate: Normal-type moves used by this pokémon are converted to fairy-type moves instead and receive a boost to their attack power. Wow. Put Hyper Beam on a Sylveon with Pixilate equipped and you can utterly maul most opponents, super effective or not. Pixilate is a hidden ability.
Sylveon has the potential to be a solid wall, particularly if you have one with Pixilate. Moonblast will be your primary weapon, and you can use it to shred the majority of the competition. Beyond that you'll need moves which can ward off poison- and steel-types; you unfortunately don't have a ton of choices, so Hyper Beam, Psyshock, or Shadow Ball will have to do. Toss in a support move like Draining Kiss to keep Sylveon alive and you have your wall.
The alternative, one that's much stronger, is to turn Sylveon into an extremely-irritating staller and Toxic user. To support Toxic you'll want a variety of moves that will keep Sylveon's HP high. Draining Kiss will do STAB damage while absorbing ample amounts of health, Protect does just that, and Light Screen / Reflect will dull the effects of attacking moves. Reflect is the better choice, though it depends more on your EV training. If you want a heftier STAB move you can go with Moonblast instead of Draining Kiss, but it's wiser to keep Sylveon alive instead. To supplement its survival skills, toss on Leftovers.
EV / Super Training
Begin with a smattering of HP. Sylveon's Defense could use work, but you're better off maxing out HP instead. Sylveon starts off with a ridiculous amount of Special Defense, though it can use at least 100 EVs in that area. Plop the rest into Special Attack. Alternatively, if you don't plan on using STAB moves or if you don't want a wall in general, pump all of your EVs into HP and Special Defense, almost ignoring Special Attack completely.
Catching a Sylveon
Most of the work in getting your Sylveon is done in Pokémon Amie. To get the Eevee, on the other hand, you'll have to do some hunting, and the only place to find Eevees in Pokémon X and Y is on Route 10. They're rare, but they're there. Check yellow flowers for a better chance of finding one.