Coordinate systems are used to uniquely define different points in space(sometimes not so as in the case of origin described by spherical or cylindrical systems).Whereas, a vector space is a set containing objects(vectors), which satisfy certain axioms.A basis vector set is any subset of the Space, whose linear combination of its elements describes the entire space.
When combining these ideas to describe space in cartesian system, the basis is fixed, while in other orthogonal systems they are position dependent. This sense of direction makes sense when using bases to describe vector fields, but when they are position dependent, does it mean our basis vector is a different set at every position in space ,as the "direction is changing",or is this notion of no significance when talking in terms of sets?
Also , I want to have some intuition regarding the relation between coefficients of the basis vectors we choose and the coordinates.
A basis $B$ of a vector space $V$ is a linearly independent subset of $V$ that spans $V$. By definition, you can write every vector $v \in V$ as a linear combination of the basis vectors. To define a coordinate system, you must have an ordering on the basis vectors, and the coordinates are defined by the coefficients.
For example, consider the basis $B = \{b_1, \cdots, b_n\}$. We can write any vector $v \in V$ as $$v = \alpha_1 v_1 + \cdots + \alpha_n b_n$$ where $\alpha_1, \cdots, \alpha_n$ are the referred to as coordinates. Keep in mind, the ordering of the basis is important. So, by definition, an ordered basis defines a coordinate system.
Now, if a coordinate system A is translated with respect to another coordinates system B, then the basis alone is not adequate for capturing this information. In such cases, you must define the origin of coordinate system A with respect to coordinates system B (or the other way around). In most applications (in my experience), we describe such relations using transformation matrices. I suggest looking at the Special Orthogonal Group $SO(n)$ (comprises of rotations) and Special Euclidean Group $SE(n)$ (comprises of rotations and translations).