physical interpretation or practical meaning of a distance

Question

How can I find/understand the physical explanation or practical meaning of a distance? For example Euclidean distance is the shortest path between two points, how about for other distances?

Answer 1

Imagine a plane of Cartesian coordinates. Now consider two different points on it.

Now tell me in how many ways you can join them??

So Euclidean distance is defined as the shortest distance between two points which is a straight line.

Answer 2

A distance (between two points, say) does not have to be a straight line or a shortest path -- a distance function is what you define it to be, but it has to be well-defined and unambiguous. For example, in the Cartesian plane, the Euclidean distance between two points is defined as the shortest path joining them, but the Taxicab distance between them is the shortest path if you 'move' from one to the other only along the 'roads' or lines parallel to the axes.