There are four Acts, or chapters, as you play through the game. The end of one act and the start of the next is loosely triggered by some action that you take.
However, you are not outright forced to complete one act before another opens. It is possible to have, for example, acts II and III both in progress at the same time, even if you have not fully completed act II.
The first act opens with a tutorial that lets new players learn the game and grasp the basic mechanics. Sir Hector, not an actual character in-game but the voice-over of the campaign and father-in-law of the king himself, explains how your story begins: with Excalibur in your hands, you are destined to reign over Britain. You're joined by some volunteers and Sir Kay, your brother-in-law, that you have to guide against some rebels in Cornwall. The first missions you encounter are in Dumnonia: you have to decide whether to help King Mark, a Righteous Knight, or King Idres of Dorset, then you have to visit Sir Gareth, who claims he has a message for you, and then have your first encounter with the magic creatures unleashed by the power of Excalibur.
It is worth noting that at points, at least on Beginner and Easy level, you can simply pass time by clicking the End Turn icon, gaining Gold, Food and experience for your Knights (by stationing them at specified Locations on the map).
You also start getting random quests and disasters which can greatly affect the flow of the game for you, depending on how you play.
The third chapter focuses on the search for the Knights of the Round Table. Most of them can be recruited in the many missions of this act, like Sir Lancelot or Mordred. It is the least influential Act in the story of the whole campaign, yet provides the player with a noticeable amount of missions and experience that are very useful in the last chapter.
The fourth and last chapter has its centre in the fight between Christians and Old Faith, represented by the Sidhe faeries. Its last and ultimate objective is to get the Graal, meaning the power over the mortal realm. It can be achieved in three different ways: with a Christian faith (Rightful or Tyrant), Dark Old Faith (Old Faith + Tyrant morality) or Light Old Faith (Old Faith + rightful morality).