The essence of geospatial analysis is discovering the relationships of objects on the Earth. Items which are closer together tend to have a stronger relationship than those which are farther apart. This concept is known as Tobler's First Law of Geography. Therefore, measuring distance is a critical function of geospatial analysis.
As you have learned, every map is a model of the Earth, and they are all wrong to some degree. For this reason, measuring accurate distance between two points on the Earth while sitting in front of a computer is impossible. Even professional land surveyors who go out in the field with both traditional sighting equipment and very precise GPS equipment fail to account for every anomaly on the Earth's surface between point A and point B. So, in order to measure distance, we must look at what we are measuring, how much we are measuring, and how much accuracy we need.
There are three models of the Earth we can use to calculate distance: