I am currently learning about tensors and many questions came with it. I started questioning my understanding of concepts such as coordinates and coordinates systems, as well as vectors, it's components and basis vectors. At first coordinates and components seemed the same concept but now I don't think they are. As much as I know (in the case of $\mathbb{R}^n$), coordinates are related to points (points being $(p_1,p_2,...,p_n)\in\mathbb{R}^n$), and components to vectors and it's corresponding basis (which can be different for different points depending on the chosen coordinate system (for example in polar coordinates)).
That being said I am still confused on the whole covariant and contravariant coordinates of a vector because I associate coordinates to what I said earlier. In my language it would be covariant and contravariant components of a vector not coordinates, but then it might just be a terminology problem. By the way can there be a coordinate system without basis vector or basis vectors without coordinate system?
With regard to the covariant and contravariant components of a vector, is it right to say that this are just useful special cases of a choice of local basis vectors (because of the dependencie of the coordinate system) when a particular change of coordinates is aplied (sort of a useful way of describing vectors when a transformation of the coordinate system is present)? Moreover is it meaningful to talk about tensors without a change of coordinates? (when I talk about tensors I am usually visualizing rank 1 tensors but I suppose it applies to the other ranks)
Summarizing I think some misconception is present on the whole coordinate system and basis vectors concepts, their differences and equalities.