Game time passes as the player performs two actions: travelling and sleeping. As the Tutorial says:
The day is divided in 8 slots. Travelling consumes this slots in different ways. Here are some examples:
Returning to your Homecamp never takes away time.
Visiting a domain takes 2 slots (or a quarter moon) (some requires 3 slots).
Exploring a biome takes 4 slots (or half moon).
Scouting for a yet unknown location also takes 4 slots.
Travelling[]
The game time used up for travelling is displayed in the destination selection screen by a moon shape. It is partially illuminated to reflect the amount of time needed. In one day, the player could choose to travel several times until the 8 slots (or a "full moon") are used up. If you try to embark on a journey that would take more time than is left at that day, you are told:
Arriving at the destination location triggers time-flow from the time at the start of the journey to the same time plus the amount needed for the journey. The flow can be observed by watching the sundial at the lower left of the screen. Upon arrival, the sun will start to move clockwise until is has arrived at its goal and stay there (e.g. one quarter of daylight time later).
As such, the player can take all the time it wishes once she/he arrived at the destination. Running instead of walking will only save some of the player's real time, not the character's game time, and it will slowly consume the character's stamina.
Night[]
At the end of the day, it will eventually get dark. To interact with your environment other than picking up or dropping items or kindling your camp fire, a light source is required. Although it has no impact on the amount of time that will have passed at the end of the day, it can make sense to "outrun the sun", i.e. to run and hurry with your plans before it gets dark.