Questing in Brave
Quest is the main campaign where you experience the story and get your gems from. I will break up structure of the worlds in these terms
Mission > Level > Sub World > World
Explanation as follows:
Mission - Makes up a Level. The number of missions will vary and that will make up 1 level
Level - Contains multiple missions. Each Sub World is made up many Levels
Sub World - Parts of a world.
World - If it is listed after tapping "World", then its a world
Progression in quest is simply beat missions, to clear levels, and finish the sub worlds to finish the world. Quest progression is important because a lot of content is locked behind it. I know it is boring and the story is kinda bland at first. It does get better in the next arc but if you are not interested, feel free to just auto and do something else while questing.
After you complete a sub world, an area with a red border may unlock. This is called an EX-Mission, using my terminology. This mission has a boss who is quite hard depending on your current units. Most of these have been power-crept but if you are working with weaker units, then they might be difficult. Beating these missions grants a gem and sometimes special rewards for other purposes.
Always remember that you get one gem every time you clear a Level.