A reference frame is any coordinate system or just a set of Cartesian axes? In Physics the idea of a reference frame is one important idea. In many texts I've seem, a reference frame is not defined explicitly, but rather there seems to be one implicit definition that a reference frame is just a certain cartesian coordinate system with a known origin.
In that case when we usually talk about two frames $S$ and $S'$ with $S'$ moving with velocity $v$ with respect to $S$ along some direction of the frame $S$ we mean that we have two sets of Cartesian coordinates $(x^1,\dots, x^n)$ and $(y^1,\dots,y^n)$ and that the coordinates are related in a time-dependent manner, for example
$$\begin{cases}y^1 = x^1 + vt, \\ y^i = x^i & \forall i \neq 1\end{cases}$$
On Wikipedia's page on the other hand we find this:

In physics, a frame of reference (or reference frame) may refer to a coordinate system used to represent and measure properties of objects, such as their position and orientation, at different moments of time. It may also refer to a set of axes used for such representation.

So that a reference frame may be a coordinate system (now, since we are not talking about axes, it could even be spherical or polar) or the axes themselves.
So what really is a reference frame? Is it just a set of cartesian axes in euclidean space $\mathbb{R}^n$? Or can it really be any set of coordinates like spherical and polar (or even another ones on more general manifolds)?
Also, how can we understand intuitively the idea of a reference frame and how this relates to the actual mathematical point of view?
EDIT: from a mathematical standpoint, a coordinate system on a subset $U$ of a smooth manifold $M$ is a homeomorphism $x: U\to \mathbb{R}^n$. The books lead me to believe that a reference frame would be equivalent to this idea. There are however, some problems in this approach:


*

*Books usually talk about reference frames just in $\mathbb{R}^n$ making the tacit assumption that the coordinates are cartesian and relating the frame with the axes. If the space is not $\mathbb{R}^n$, in truth, Cartesian coordinates aren't even possible and will problably be curved.

*Reference frames are present in Newtonian Mechanics, so it should be possible to define them without resorting to the notion of spacetime.

*Coordinate system are ways to assign tuples of numbers to points. But a reference frame can move around, something I think a coordinate system as defined in mathematics can't.
These three points are the at the core of my doubt. Reference frames shouldn't need anything spacetime related to be defined since they are present outside of relativity. Coordinate system as defined in mathematics cannot move around, so that reference frames shouldn't be synonimous to coordinate systems. And finally, if the space is not Euclidean, Cartesian axes are not possible.
So based on this, what's really a reference frame?
 A: Let $M$ be your spacetime, a smooth manifold equipped with (pseudo) Riemannian metric (for example $\mathbb{R}^{(1,3)}$ for special relativity). 
The set of reference frames is the frame bundle over $M$, usually denoted $FM$.  Explicitly a frame at point $p$ in $M$ can be viewed as an ordered orthonormal basis (with respect to the the inner product defined by the metric) for the tangent space at $p$, $T_pM$. 
For example, in metrics with Lorentzian signature in dimension 4, these frames are related by rotations in $\mathbb{R}^{(1,3)}$, aka Lorentz transformations, as expected.
A: The most direct and beautiful answer to this important question has been provided by Taylor and Wheeler in their famous book "Spacetime Physics". If you google "spacetime physics wheeler frame of reference" for images, you will be led to a picture of space divided regularly into a 3-dimensional grid pattern. The crucial thing though is that at each grid point there is a clock. And all the clocks (in the entire universe) are in sync! [For a given frame of reference.]
This, simply, is what we mean by a "frame of reference". It means that any "event", an occurence at a point in space and time, can then be labeled by the spatial coordinates and the clock.
A: 
So based on this, what's really a reference frame?

In pre-relativistic mechanics, reference frame is a system of points whose mutual distances are assumed constant - a rigid body.
For measurements of position on Earth, the reference frame is often Earth's body, assumed to be rigid.
For measurements of position in space, Earth's body could be used.
The rigid body does not have to be one connected body, though. When rotation of the Earth is to be described, frame with origin in the Earth (Sun) with axes pointing to distant stars can be used as a reference frame.
A: Based on Einstein's assertion:

All our well-substantiated space-time propositions amount to the determination of space-time coincidences {such as} encounters between two or more material points. 

the notion "reference frame" should likewise be expressed in terms of (requirements on) "material points" and "space-time coincidences" in which they did, or didn't, take part.
A suitable general definition then seems to be 


*

*a set of "material points" such that no pair of them had ever been coincident. In some contexts a set with this property is called "timelike congruence". Moreover it may be required that

*for any three such "material points", $H$, $J$, $K$, and for each event $\varepsilon_{HP}$ (in which $H$ took part, along with some suitable participant $P$ who didn't belong together with $H$ to a timelike congruence)    


*

*either $H$ saw that $J$ had seen event $\varepsilon_{HP}$, and even before that $H$ saw that $K$ had seen event $\varepsilon_{HP}$;

*or $H$ saw that $K$ had seen event $\varepsilon_{HP}$, and even before that $H$ saw that $J$ had seen event $\varepsilon_{HP}$; 

*or $H$ saw that $J$ had seen event $\varepsilon_{HP}$, and in coincidence $H$ saw that $K$ had seen event $\varepsilon_{HP}$,
such that all members of the reference frame maintain (mutual, triple-wise) reference between each other by "pings".
Finally, there may be additional requirements, for instance related to "rigidity" (of "ping relations" between any three members), or (the existence or non-existence of) "gaps" or other "topological properties" (as far as they are derived from coincidence determinations).     
Any (subsequent) one-to-one assignments of coordinate tuples to the individual members of a reference frame, and to the events in which the individual members took part (separately) are of course equivalent, and arbitrary, and without any further meaning by themselves. But any one such coordinate assignment may or may not represent the given geometric or topological relations between reference frame members, the sequence of events in which any one member had taken part, etc.
A: It would help to read this section of Galilean Invariance as this reconciles it nicely with more intuitive notions of relative frames of reference.
Two observers moving at different speeds (or help us all accelerations), would not agree on the simultaneity of some events. This represents a shift in relative time due to motion, which is why you see a $vt$ component turn up in the math.
Reference frames do not belong to any one object, I feel. Rather, they are ascribed to velocities in spacetime, and have transforms to convert between the two (eg Lorentz boost). What they represent is distinct notions of simultaneity.
So in one, I could say reference frames are comparative representations of simultaneity for inertial observers.
Beyond that is wholly my opinion, as I would go so far as to say that reference frames describing the same velocity in two different regions of space at the same time, are not the same thing due to locality - this would come up when you examine acceleration horizons / event horizons etc.
On top of that it is hard to pin down if a reference frame from one second ago is truly the same one as you're in now. There may be no observable differences though to me that is insufficient as for all we know we're traveling through reference frames instead of within them / with them. (also my opinion)
