The precise notions you need are that of a manifold (and the definition of charts and atlases (this terminology obviously derived from their usual meaning in everyday English) which go along with it). For that, I strongly suggest you Frederic Schuller's lectures.
Let me start of slightly informally. By coordinates, we really mean a mapping of some kind. We take a certain point of interest, and assign it some collection of numbers. So, you could also think of this as a "numericalization" process of some sort. Slightly more explicitly, start with a set $M$ of interest. By a (local) coordinate system on $M$, we mean an injective mapping $\psi:U\to\Bbb{R}^n$ for some $n$, where $U$ is some subset of $M$. Some remarks (a lot of it is terminology/vocabulary):
- Usually, people refer to a local coordinate system as the pair of information $(U,\psi)$, so the make the domain explicit in the notation. This is somewhat redundant (but of course, it doesn't hurt to be explicit) because $\psi$ is by definition a function, and functions by definition have some domain attached to them. Some other names include "$(U,\psi)$ is a local coordinate system", or "$(U,\psi)$ is a local coordinate chart", or "$(U,\psi)$ is a local chart" and so on.
- We use the adjective "local" in "local coordinate system" because the domain of the mapping $\psi$ is merely a subset $U\subset M$. If we had $U=M$, then we would call this a global coordinate system on $M$.
- The injectivity of $\psi:U\to\Bbb{R}^n$ means of course that if you restrict the target space to the image, then you get a bijective mapping $\psi:U\to\psi[U]$, which then has an inverse $\psi^{-1}:\psi[U]\to U$.
- The idea of this definition is that the object $\psi$ (a function) assigns for each point $p\in U$ a certain tuple of numbers $\psi(p)\in \Bbb{R}^n$. We call the tuple $\psi(p)$ "the coordinates of the point $p$, relative to the coordinate system $(U,\psi)$". Now, by injectivity, if you give me any tuple of numbers $(a^1,\dots, a^n)\in \psi[U]$, then I can tell you which point in $U$ this tuple corresponds to, namely $\psi^{-1}(a^1,\dots, a^n)$.
- Now, since $\psi$ is a mapping which has target space $\Bbb{R}^n$, we can think of it as being comprised of $n$ different mappings $\psi=(\psi^1,\dots,\psi^n):U\to\Bbb{R}^n$. So, $\psi^1:U\to\Bbb{R}$, $\psi^2:U\to\Bbb{R},\dots, \psi^n:U\to\Bbb{R}$ are all certain functions which fit together to give an injective mapping $\psi=(\psi^1,\dots, \psi^n):U\to\Bbb{R}^n$. We call $\psi^i$ the $i^{th}$ coordinate function of the coordinate system $(U,\psi)$.
- Given a point $p\in U$, I said above that $\psi(p)$ is called the coordinates of the point $p$ with respect to the coordinate system $(U,\psi)$. We also say that the number $\psi^1(p)$ is the first coordinate of the point $p$ with respect to the coordinate system $(U,\psi)$. Similarly, for any integer $1\leq i \leq n$, we say the number $\psi^i(p)$ is the $i^{th}$ coordinate of the point $p$ relative to the coordinate system $(U,\psi)$.
- In this subject of differential geometry, it is tradition to not write the coordinate functions as $\psi^1,\dots, \psi^n$. Rather, we use the notation $x_{\psi}^i$ to denote the function $\psi^i$. So, we have $\psi=(\psi^1,\dots, \psi^n)\equiv (x_{\psi}^1,\dots, x_{\psi}^n)$. Finally, mathematicians are a lazy bunch so sometimes they may drop the subscript $\psi$ all together, and just say $\psi=(x^1,\dots, x^n)$ is a local coordinate system.
- Finally I should mention that in practice, it is easier to specify the map $\psi$ in the opposite direction, meaning it's easier to give a definition for the inverse. So, sometimes, it is easier to start with some set $A\subset\Bbb{R}^n$ and define an injective mapping $\tilde{\psi}:A\to M$ (this is usually called a local parametrization of $M$; here the adjective "local" refers to the fact that the image of $\tilde{\psi}$ need not be all of $M$, i.e it need not be a surjective mapping). Then, we set $U=\tilde{\psi}[A]$, and define $\psi=(\tilde{\psi}^{-1}):U\to A\subset\Bbb{R}^n$. This $(U,\psi)$ is now a local coordinate system on M$.
Notice how in this preliminary definition, I made no reference at all to any sort of linear algebra/vector space structure. I didn't talk about basis vectors (linear algebra), I didn't talk about orthogonality (geometry in linear algebra). None of that. Coordinates should really just be thought of as a process of assigning tuples of numbers to points in a certain set.
For the sake of completeness, let me mention what I did not discuss above. The definition above is incomplete because I didn't really discuss the role of the integer $n$. In order to avoid pathologies, we should impose certain conditions on the set $M$ (namely that it is a topological space of some kind) and that all the maps are homeomorphisms and so on (because it is a simple fact in set theory that $\Bbb{R}\cong \Bbb{R}^n$ are sets of the same cardinality, so there is a bijection between $\Bbb{R}^n$ and $\Bbb{R}^m$ for any $n$ and $m$). This is all done more systematically in the lecture series I linked to above. My purpose in this answer is to tell you roughly what coordinates are, and to really emphasize what they are not.
Finally, I should mention that we can have more than one coordinate system on the same set $M$, and we can also have more than one coordinate system with the same domain $U$, such as $(U,\psi), (U,\tilde{\psi})$, where $\psi,\tilde{\psi}:U\to\Bbb{R}^n$ are two completely different maps. So far I have talked a lot about the abstract definition, but how does it relate to some familiar examples?
Example 1: Vector spaces, bases, and associated coordinate system.
Let $V$ be an $n$-dimensional vector space over $\Bbb{R}$. Let $\beta=\{v_1,\dots, v_n\}$ be a basis for this vector space (linearly independent and spanning). As a result, for each vector $v\in V$, there exist unique scalars $a^1,\dots, a^n\in\Bbb{R}$ such that
\begin{align}
v&=\sum_{i=1}^na^iv_i.
\end{align}
We now define a global coordinate system $\psi:V\to\Bbb{R}^n$ as $\psi(v)= (a^1,\dots, a^n)$, where the $a^1,\dots, a^n$ are the unique numbers making the equation above true.
Perhaps a slightly easier way of describing this is that we first consider the mapping $\Bbb{R}^n\to V$ defined as $(a^1,\dots, a^n)\mapsto\sum_{i=1}^na^iv_i$. Now, because $\beta=\{v_1,\dots, v_n\}$ is a basis for the vector space, it follows the mapping is both injective (due to linear independence) and surjective (due to spanning). Hence, it is bijective, and thus has an inverse. This inverse mapping what I called $\psi:V\to\Bbb{R}^n$ above.
I should really write $\psi_{\beta}$ to indicate that this mapping (actually a linear isomorphism) has been constructed from the basis $\beta$. So, it is in this sense that a basis for a vector space gives rise to a coordinate system in the above sense. This is the "natural" way of getting a coordinate system using a basis because it fully incorporates the linear structure on the space $V$. Conversely, given any isomorphism $\psi:V\to \Bbb{R}^n$, we get an associated basis $\beta$ of $V$, namely $\beta=\{\psi^{-1}(e_1),\dots, \psi^{-1}(e_n)\}$, where $e_i=(0,\dots, 1,\dots 0)\in\Bbb{R}^n$ is the vector with a $1$ in $i^{th}$ spot and $0$ elsewhere.
So, in the setting of vector spaces, there is a one-to-one correspondence between linear isomorphisms $\psi:V\to \Bbb{R}^n$ (which qualify as a "coordinate system on $V$") and bases $\beta$ of $V$. So, this (and the fact that in introductory math you usually only deal with vector spaces like $\Bbb{R},\Bbb{R}^2,\Bbb{R}^3$) is why people may initially just define a coordinate system to be a basis. However, I would caution against such an identification; you should keep these notions separate: a coordinate system is different from a basis.
Example 2: $\Bbb{R}^n$
The set $M=\Bbb{R}^n$ is rather special because it appears in the very definition of a coordinate system by virtue of being the target space of $\psi$. So, a global coordinate system on $M=\Bbb{R}^n$ is really just an injective mapping $\psi:\Bbb{R}^n\to\Bbb{R}^n$. There is a very obvious example of such a mapping, namely the identity function: send each point $p\in\Bbb{R}^n$ (which by definition is an $n$-tuple of numbers) to the same $n$-tuple $p$, i.e $\psi(p):=p=(p^1,\dots, p^n)$. So, $(\Bbb{R}^n\text{id}_{\Bbb{R}^n})$ qualifies as a coordinate system on $\Bbb{R}^n$, and it is this special coordinate system that we refer to as "Cartesian coordinates". It is of course standard notation to denote the coordinate functions $\psi^i=(\text{id}_{\Bbb{R}^n})^i$ as $x^i$. So, Cartesian coordinates are $(x^1,\dots, x^n)$, and in lower dimensions we simply use the notation $x$ (if $n=1$), $(x,y)$ if $n=2$, and $(x,y,z)$ if $n=3$ to denote these coordinate functions.
Also, if you think of $M=V=\Bbb{R}^n$ as a vector space, and consider the "standard basis" $\beta=\{e_1,\dots, e_n\}$, then as described in the previous example, this gives rise to a coordinate system $\psi:\Bbb{R}^n\to\Bbb{R}^n$ (which turns out to be a linear isomorphism), and since we chose specifically the basis $\{e_1,\dots, e_n\}$, this mapping $\psi$ is equal to the identity map, i.e we recover "Cartesian coordinates". It also turns out that if you choose the standard inner product on $\Bbb{R}^n$, that this basis is orthonormal.
It is because of these many ways of viewing things that it is often so difficult (in the beginning) to separate the notions of coordinates, basis vectors, Cartesian coordinates, and the notion of orthonormality and so all these notions get jumbled up into one big idea. Hopefully this clarifies the distinction between these objects.
Different Coordinates on the Same Space: Polar Coordinates
As I mentioned previously, if you consider one and the same set $M$, you can have many many different coordinate systems. Let us now introduce polar coordinates in a precise manner. Let $M=\Bbb{R}^3$. Think of this $M$ as the abstract set on which we wish to introduce a polar coordinate system. Now, consider the set $A=(0,\infty)\times (0,\pi)\times (0,2\pi)\subset\Bbb{R}^3$. And now consider the mapping $\tilde{\psi}:A\to M$ defined as
\begin{align}
\tilde{\psi}(r,\theta,\phi)= (r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta).
\end{align}
Because of how the domain $A$ is defined, it is straight-forward to show that this mapping is injective. We call the image of this mapping $U:=\tilde{\psi}[A]$, and we let $\psi=(\tilde{\psi})^{-1}:U\to A\subset\Bbb{R}^3$. It is the pair $(U,\psi)$ that we call "spherical polar coordinates in $\Bbb{R}^3$" (though we should really say "spherical polar coordinates on the set $U\subset\Bbb{R}^3$" to emphasize that the coordinate system is only defined on a proper subset $U$ of $\Bbb{R}^3$).
- Remember how for any coordinate system we can consider the coordinate functions? We can do the same here, so we can consider the spherical polar coordinate functions $\psi=(\psi^1,\psi^2,\psi^3)\equiv (x_{\psi}^1,x_{\psi}^2,x_{\psi}^3):U\to A=(0,\infty)\times (0,\pi)\times (0,2\pi)$. It is now of course tradition to abscond with the letter $\psi$ and this extra notation, and write these functions as $r\equiv x^1_{\psi}, \theta\equiv x_{\psi}^2,\phi\equiv x_{\psi}^3$. So, we have three functions $r:U\to (0,\infty), \theta:U\to (0,\pi), \phi:U\to (0,2\pi)$. For each point $p\in U\subset M=\Bbb{R}^3$ we can thus associate a 3-tuple of numbers $(r(p), \theta(p),\phi(p))\in A\subset\Bbb{R}^3$, where the geometric meaning is of course that we assign to the point $p$ its distance from the origin, and the two angles measured appropriately.
Now, the key point is that polar coordinates are a coordinate system on the region $U\subset M=\Bbb{R}^3$. Now, recall the definition of Cartesian coordinates from the previous example. By definition, they make up the identity map $\text{id}_{\Bbb{R}^3}=(x,y,z):\Bbb{R}^3\to\Bbb{R}^3$. Now, by restricting the domain of these maps, we of course get an injective map $(x,y,z):U\to\Bbb{R}^3$, so this is still a coordinate system on $U$.
So, we now have two different coordinate systems on the same set $U\subset M$. One is Cartesian coordinates $(U,(x,y,z))$, and the other is polar coordinates $(U,(r,\theta,\phi))$. The relationship between these functions is that for each $p=(p^1,p^2,p^3)\in U$, we have
\begin{align}
\begin{cases}
x(p)=p^1=r(p)\sin\theta(p)\cos\phi(p)\\
y(p)=p^2=r(p)\sin\theta(p)\sin\phi(p)\\
z(p)=p^3=r(p)\cos\theta(p)
\end{cases}
\end{align}
Since these equalities hold for all $p\in U$, we have that
\begin{align}
\begin{cases}
x&=r\cdot (\sin \circ \theta)\cdot (\cos\circ \phi)\\
y&=r\cdot (\sin \circ \theta)\cdot (\sin\circ \phi)\\
z&=r\cdot (\cos \circ \theta)
\end{cases}
\end{align}
This is a proper way of expressing the relationship between the coordinate functions. However, mathematicians are obviously a little lazy, so we suppress the composition symbol and simply write that "spherical polar coordinates are defined (implicitly) by the relations"
\begin{align}
\begin{cases}
x&=r\sin\theta\cos\phi\\
y&=r\sin\theta\sin\phi\\
z&=r\cos\theta
\end{cases}\tag{$*$}
\end{align}
Hopefully this example allows you to reconcile the formal/abstract definition of a coordinate system which I gave initially, with the usual language with which it is typically introduced. Also, hopefully this example illustrates why it is sometimes easier to define a parametrization first (i.e the inverse of the coordinate system).
Final Example: Parabolic Coordinates
One typically only encounters Cartesian, polar and cylindrical coordinates, so let me instead give you a different example to illustrate how the formal definition matches up with how things are done in practice. We shall consider Parabolic Coordinates in part of $\Bbb{R}^2$. If you see the first thing which Wikipedia says:
Two-dimensional parabolic coordinates $(\sigma,\tau)$ are defined by the equations, in terms of cartesian coordinates:
\begin{align}
\begin{cases}
x&=\sigma\tau\\
y&=\frac{1}{2}(\tau^2-\sigma^2)
\end{cases}\tag{$**$}
\end{align}
then it may not be clear (a-priori) how this corresponds to the above formal definition I gave for a coordinate system as an injective mapping from some set. WHat are the $M,U,\psi$ here? Well, as you may have hopefully already picked up from the previous examples, it is sometimes easier to provide an implicit definition for the coordinate system in terms of another already well-defined and well-understood coordinate system (i.e defining $\tilde{\psi}:A\to M$, the parametrization is sometimes simpler than defining its inverse $\psi=(\tilde{\psi})^{-1}:U=\tilde{\psi}[A]\to A$). Once we provide the definition of $\tilde{\psi}$, it is then much easier to use abstract tools like the inverse function theorem to prove that it is indeed locally injective/ can be locally inverted.
Anyway, here $M=\Bbb{R}^2$, it is a little annoying to work out what exactly $U,\psi$ are. Anyway, the point I wish to drive home is that these parabolic coordinates, like any other coordinate system, are just a device by which we can take a point $p\in U$ and obtain a tuple of numbers $(\sigma(p),\tau(p))$. The relationship between these coordinates and the usual Cartesian coordinates $x(p):=p^1, y(p)=p^2$ is then given by equation $(**)$ applied to each point $p\in U$.
What is going on with the Pictures
Typically, if we wish to visualize a coordinate system $(U,\psi)$, which has the coordinate functions $(x_{\psi}^1,\dots, x_{\psi}^n)$, what we do is we look at the set of points where each $x_{\psi}^i$ is constant. So, for each number $t\in\Bbb{R}$, we like to look at the set of points
\begin{align}
\{p\in U\,:\, x_{\psi}^i(p)=t\}\qquad (1\leq i \leq n)
\end{align}
So, for example, in $M=\Bbb{R}^2$, the set of points with constant $x$-coordinate are "vertical lines", while the set of points with constant $y$-coordinate are "horizontal lines", the set of points with constant $r$-coordinate are circles centered at the origin, and set of points of constant $\theta$-coordinate are half-rays. In the parabolic coordinate system, the set of points with constant $\sigma,\tau$ are parabolas of some kind and so on.
So, whenever one has a coordinate system $(U,\psi)$ in mind and wishes to draw pictures, what we do is we look at everything in the image of this mapping, i.e we draw things in $\psi[U]\subset \Bbb{R}^n$. Since the image is a subset of $\Bbb{R}^n$, it is perfectly fine to draw an $n$-dimensional grid as usual, and label to axes as $x_{\psi}^1,\dots x_{\psi}^n$. So, a tuple $(a^1,\dots, a^n)\in\psi[U]$ then obviously just corresponds to a point $p\in U$ with coordinates $x_{\psi}^1(p)=a^1,\dots, x_{\psi}^n(p)=a^n$.
Bottom line: coordinates are not the same thing as a basis, they are not the same thing as a orthogonal basis. None of that. Yes, given a basis for a vector space, we can define a global coordinate system, but don't let this simple fact lead you to mixing up the two notions. Coordinate systems are just a systematic way of assigning numerical quantities to certain points of interest (and this had better be injective so that two different points do not get assigned the same tuple of numbers). Finally, pictures that we draw are usually represented in the target space of the mapping $\psi$ (we do this because the target space is $\Bbb{R}^n$, and we have developed our intuition from $\Bbb{R}^n$).
