Pokemon X and Y's only evolution that comes from a previously released Pokemon is Sylveon, the newest Eeveelution. But with Eevee already having a ton of different evolutions that take advantage of nearly every element of the Pokemon games (different stones, different times of day, different moods), the actual means of turning Eevee into Sylveon were unknown. Many speculated it would involve Moon Stones, or leveling up an Eevee with a different Fairy-type Pokemon, or something else, but after spending a few dozen hours in the game we have the solution--as well as a video that shows exactly how to do it.
First, you'll need to teach your Eevee a fairy-type move, but that's not enough! You also need to make it very, very happy. The solution lies in the new Pokemon-Amie system, which allows you to play and pet your Pokemon. In order to evolve to Sylveon, you're going to need to have all three sections in Amie--Affection, Fullness, and Enjoyment--up to max. Filling up Enjoyment is as easy as playing a bunch of minigames, and that will give you enough snacks to make sure your Pokemon's Fullness is max.
Once you've maxed out all three of these teach your Eevee a fairy move, level up, and--bam! You're the proud owner of a shiny new (well, not actually shiny, most likely) Sylveon! Take note to make sure your Fullness is maxed going into battle, since it can go down over time (or between battles), but as long as you follow these instructions you'll have a Sylveon after about an hour or two's work!