I am currently studying computer graphics and OpenGL on my own and reading about tesselation. In the book OpenGL SuperBible by G. Sellers, the following is said: "Tesselation in OpenGL works by breaking down high-order surfaces known as patches into points, lines, or triangles." To be honest, the term high-order surface is too abstract for me.
Hence my question: What is the rule to qualify a geometry as a high-order surface?
A "higher-order surface" is a (conceptual) surface which will be transformed into a more refined version of itself (the "lower-order surface"). If you have 4 points, you could create a more refined set of points by putting points on the lines between them to form a quadrilateral made of points. Or you could put points along curves that smoothly go between the 4 points. There are many ways to generate points "between" the 4 points. Your final set of points may not even include the original 4 points.
The 4 points are the "higher-order surface"; the points you draw based on them are the "lower-order surface".
Tessellation is the process of taking that "higher-order surface" and generating the "lower-order surface" from it, based on some algorithm that you provide.