In the above figure, the vertices expressed as blue dots are "equivalent-looking." Although my expression is somewhat ambiguous, I believe one can simply answer it. How can we call such vertices? What is the relation between blue vertices? How can I properly express the relation between the green vertices graph-theoretically? As general as possible...
It seems likely that you are looking for the property v is equivalent to w if their is a graph automorphism f such that f(v)=w. A graph automorphism being a bijection of vertices preserving edge relations. i.e. (v,w) is an edge if and only if (f(v),f(w)) is an edge. A graph would be vertex transitive if all vertices were equivalent in this way. This seems to capture the idea of a graph "looking the same" whether viewed from the vertex v or the vertex w.