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I'm not sure, whether we discuss SR or GR.
In GR we deal with a differentiable pseudo-Riemannian manifold. There coordinates are usually defined on an open subset of the manifold (a socalled map). Then you have several maps covering the entire manifold. On the overlaps between any two maps there's a (local) diffeomorphism between the corresponding coordinates. All maps together build an atlas. Anything physical (or geometrical) is independent of the choice of maps and atlasses though and thus defined via invariant properties of tensors and tensor fields.
As @PeterDonis said above, the coordinates defined on a map always imply (local) reference frames, namely a set of basis vectors in the tangent space at any point covered by the map defined by the coordinate lines through this point. The most simple way to describe this reference frame is to use the coordinates of the map and the holonomous basis and co-basis vectors and the tensor components with respect to this holonomous basis. This is what's usually taught first in textbooks like Landau and Lifshitz vol. 2, and you get very far with this description in SR.
Sometimes you however need more sophisticated tools as the concept of frame fields also described by @PeterDonis above. There at any point you can define a time-like worldline through this point and construct a pseudo-orthonormal set of (tangent) basis vectors. This you need, e.g., to define spinors and spinor fields in GR. In SR that's not a big issue, because there you usually work with a global orthonormal basis defining at the same time a global Galilean reference frame.
In GR we deal with a differentiable pseudo-Riemannian manifold. There coordinates are usually defined on an open subset of the manifold (a socalled map). Then you have several maps covering the entire manifold. On the overlaps between any two maps there's a (local) diffeomorphism between the corresponding coordinates. All maps together build an atlas. Anything physical (or geometrical) is independent of the choice of maps and atlasses though and thus defined via invariant properties of tensors and tensor fields.
As @PeterDonis said above, the coordinates defined on a map always imply (local) reference frames, namely a set of basis vectors in the tangent space at any point covered by the map defined by the coordinate lines through this point. The most simple way to describe this reference frame is to use the coordinates of the map and the holonomous basis and co-basis vectors and the tensor components with respect to this holonomous basis. This is what's usually taught first in textbooks like Landau and Lifshitz vol. 2, and you get very far with this description in SR.
Sometimes you however need more sophisticated tools as the concept of frame fields also described by @PeterDonis above. There at any point you can define a time-like worldline through this point and construct a pseudo-orthonormal set of (tangent) basis vectors. This you need, e.g., to define spinors and spinor fields in GR. In SR that's not a big issue, because there you usually work with a global orthonormal basis defining at the same time a global Galilean reference frame.