Let's say we are in $\mathbb{R}^n$ and someone says let $(x_1,\ldots, x_n)$ be a coordinate system. What does this precisely mean?
A partial answer would be to say that this means that there exists a basis of $n$ vectors $e_1,\ldots, e_n$ and the point $(x_1,\ldots, x_n)$ is just shorthand notation for the point $x_1e_1+\ldots +x_ne_n$. This clearly is too prohibitive since, for example, it does not allow for polar coordinates. So what is it then?
Polar coordinates are generally (strictly speaking) not a coordinate system for $\mathbb{R}^n$; at best they are coordinate systems for $\mathbb{R}^n \setminus \{0\}$.
Since you tagged differential geometry: a coordinate system for $\mathbb{R}^n$ is a collection of $n$ functions $x_1, \ldots, x_n :\mathbb{R}^n \to \mathbb{R}$ such that at every point $p\in \mathbb{R}^n$, their differentials $dx_1, \ldots, dx_n$ form a basis of $T_p^* \mathbb{R}^n$.
Depending on the context, this can be more or less restrictive; for example, sometimes it is convenient to consider more general parametrizations of a manifold (instead of by patches of $\mathbb{R}^n$, by patches of a convenient manifold; one may regard polar coordinates as the parametrization of $\mathbb{R}^n\setminus\{0\}$ by $\mathbb{R} \times \mathbb{S}^{n-1}$). Sometimes it maybe useful to require additional properties of the coordinate functions (on $\mathbb{R}^n$, one may require them to be linear functions).