As you progress though story mode you'll be able to unlock both story and horde levels.  To know more about each mode click here.

Unlock Horde Mode

To unlock horde mode you need to conquer all 3 levels of a world in story mode. 

Unlock Story Worlds

There are 8 worlds each including 3 levels. When you complete all 3 levels you'll unlock the next world with its 3 levels.

For example, Japan world will only get unlocked once you complete all 3 levels in Egypt world.