I am having a really hard time in understanding the difference between coordinate and non-coordinate basis. I tried looking for some relevant questions here but none answers my concerns, may be my question is too basic.
The way I explained myself the difference between co-ordinate and non-coordinate basis is in terms of the orthonormality of the basis vectors (I am reading text on GR, Schutz). I had understood that the difference is orthonormality i.e. coordinate basis are orthonomal while non-coordinate basis are just orthogonal. But that later the later part of the text contradicts my understanding by stating that "In textbooks that deal with vector calculus in curvilinear coordinates, almost all use the unit orthonormal basis rahter than coordinate basis.".
So now I am completely lost. Can anyone explain what is the difference between coordinate and non-coordinate basis in less mathematical terms (more in terms of the Physics or with geometric arguments that I can visualize). Or just point me to some text I can read.
Update: The answer given to this question confirms my understanding. I later came to realize that the author meant that the basis vectors are called "unit orthonormal basis rahter than coordinate basis".