Chapters are a way to quickly navigate the game map, and they show at a glance the achievement progress you've made for each chapter! You can access the chapters from the menu panel on the left side of the screen.
Each chapter represents a section of levels. For example, Chapter 1 is levels 1 to 15, Chapter 2 is levels 16 to 30, and so on. Tapping on a chapter will automatically move you to that section of levels on the map.
From the chapters menu, you can also see the achievements available for each chapter, and your progress towards completing those achievements. Once you complete an achievement, you can collect the reward from the chapters menu! Complete all the achievements in a chapter, and that whole chapter will then be marked as completed as well.