Sylveon debuted in the Pokémon anime in one of the Pikachu shorts. These are shorts that usually only feature Pokémon and often air at the start of Pokémon movies, although some have also been released on DVD or on the official Pokémon website. Sylveon's first appearance was in a short called Eevee and Friends that aired first in Japan in July 2013, and then on the official website in English in December 2013.
The short features Eevee and all the Eeveelutions who live together. Sylveon invites Pikachu and it's friends to a party. The food is cooked by Meowth and Wobbuffet, and some of the Pokémon perform acts on a stage. Afterwards, all the Pokémon follow Umbreon and find Gothitelle who uses it's psychic powers to display a beautiful view of the stars and galaxy.
This short is incredibly cute, and I particularly enjoyed seeing all the Eeveelutions together! It was also a great introduction to Sylveon, and it was especially cute to see how it uses it's ribbons to pick up and comfort the other Pokémon. Sylveon's feelers also stretch, which was unexpected.
Sylveon first appears in the main Pokémon anime in episode 13 of the XY series, called Kindergarten Chaos! This episode aired in Japan in January 2014, and in the USA in April 2014. Not only did this episode show Sylveon for the first time, but it also introduced the Fairy-type into the anime. In it we see Sylveon successfully battle Ash's Froakie, and it uses it's feelers to block attacks. Like in Eevee and Friends, Sylveon's feelers stretch.
Sylveon has since appeared more in the anime. One of Ash's companions, Serena, even has an Eevee that evolves into a Sylveon. If I ever get around to actually watching the anime again, I'll update this section with more information!