Attack Combo's or Chains are another new element to the Pokémon universe

The way to Chain the moves together first is to go to Gulpin's Shop. He will do it for you if you have found the TM that allows it to be done.
You can find the TM's anywhere, in a dungeon, in a forest...they're just sitting on the ground or are left behind by a Pokémon. You get to pick the moves to be chained such as Scratch & Leer for example.
This saves time in battle when the moves are done in one turn and can add extra damage and more attacks. However lose it with care as you can lose it...