Zones refers to any of the maps where you are actually dashing. Each zone has a set of monsters that may appear plus one rare monster and one boss at the end. They all have a path down the center that Dash will follow. Across the path may be different gates that must be destroyed in order to keep going. There are a number of non-interactable objects on the sides, and occasionally there are breakable objects with health bars that have no effect when destroyed.
Each zone has a distance that can vary greatly depending on which of the modes you are in. They are usually between 100 and 250. If they go beyond that, they are usually broken into a subset number, such as Gloom Cavern 3, or Lava Forge 16. In Classic mode and Hero mode, there will be a wooden marker with the best distance you've ever traveled before dying.
There are 6 normal zones in the regular modes: Goblin Hills, Dark Forest, Gloom Cavern, Deadly Marsh, Castle, and Dark Castle(Castle 2 in the beastiary). There are 3 others available in the Daily Dungeon: Catacombs, Sunken Temple, and Lava Forge.
Each of the mini-games uses one of the normal zones.