well, it is all about getting "hearts", see those hearts that appear when you feed a Pokemon, play with it, and pet it? Well after a certain amount, your affection goes up by one. Once your affection is maxed, it can never go down, and you can evolve eevee into sylveon. Pokemon with max affection also do special things in battle. For example, it will land critical hits easier, it will "read your mind" and dodge attacks a lot, and if it gets poisoned or paralyzed, it will shrug it off and lose the status condition, and much more. It is very useful in the normal game itself, but doesn't work when having wifi battles.