Kalos brought two major evolution changes: Mega Evolution and the Fairy type. Sylveon introduced Affection-based evolution through Pokemon-Amie. 28 existing Pokemon gained Mega forms, temporary battle transformations triggered by Mega Stones. Charizard and Mewtwo each got two different Mega forms.
Your Pokemon holds a Mega Stone (specific to its species), and you hold a Key Stone (built into the Mega Ring). During battle, select the Mega Evolution option alongside your attack. The Pokemon transforms for the rest of the battle with boosted stats and sometimes a changed type or ability. You can only Mega Evolve one Pokemon per battle.
Sylveon was the first Fairy-type Eeveelution. Raise Eevee's Affection to 2 hearts in Pokemon-Amie (pet it, feed it Pokepuffs, play minigames) and make sure it knows a Fairy-type move. Then level it up. If it knows a Fairy-type move and has enough Affection, it becomes Sylveon instead of Espeon or Umbreon.
28 Mega forms in X and Y alone. Charizard and Mewtwo each got two versions (X and Y variants), while the rest got one each. Lucario, Kangaskhan, Gengar, Blaziken, and Garchomp were some of the strongest competitively. ORAS later added 20 more Mega forms.
Scattered all over Kalos, and many only appear during the postgame between 8 PM and 9 PM. You need the upgraded Mega Ring from Sycamore. Some stones are version-exclusive: Charizardite X is in X, Charizardite Y is in Y. The time restriction means you'll miss most of them on a normal daytime playthrough unless you adjust your 3DS clock.
Beyond Sylveon, Gen 6 actually introduced very few new evolution lines. Most of the evolution focus was on Mega Evolution as a temporary battle form. Pancham evolves into Pangoro if you have a Dark-type in your party at level 32. Inkay evolves into Malamar by holding the 3DS upside down when it levels up at 30. Those two are the most notable non-Mega additions.
Only Sylveon. Affection (from Pokemon-Amie) is separate from Friendship (the hidden stat). Friendship-based evolutions like Espeon and Umbreon still use the old Friendship system, not Amie hearts. The two systems look similar but they're tracked independently. It's confusing, and X/Y doesn't explain the difference well.