The materials vary depending on the type of ship. Most common ships are made of wood with reinforced steel, but more modern ships particularly the military and speedships will be constructed mostly of steel and other stronger materials.

As for how they fly, they were historically made based of Astral engineering requiring though there have been modern advancements. I don't know the full specifics, but it's the technology and magic used in the engine that powers the ship and keeps it airborne.