Story volumes are locations on the World Map, beginning in West Kruna. They are numbered, and each contains a number of zones within it, which must be completed in order. The zones must be fought at Normal difficulty first, then Hard, and finally Nightmare. In order to advance to next area you do not necessarily have to clear all the mobs. It is enough to defeat the main boss of the zone on Normal difficulty (although you will need to clear enemies to reach the boss).
Note: Since the reboot, story volumes are separated into Legacy and Standard. In Legacy story volumes, you can use cards from any set. Standard story volumes, however, require that your deck only contain cards from the two most recent card volumes (Civil War, and Tides and Time).