Non-punctual Boundary In the book of Bill Thurston, Three dimensional geometry and topology, there is an exercise to show torus can be partitioned into 7 countries, each on one piece and has common (non-punctual) boundary with every other country. please, can anyone clarify the meaning of non-punctual ?

As Mark Bennet said, it means every two regions have a nontrivial piece of boundary in common: nontrivial means containing a topological arc, i.e., not being a point. This is a standard detail in Map coloring problems, and Thurston's exercise is in the context of map coloring.