Coordinate charts vs. coordinates on manifolds I just realised that I'm confused what coordinates really means in the context of manifolds. 
For example, say $M=S^2$. Then we can define smooth charts as follows: Let the open sets be $U = S^2$ minus the north pole and $V=S^2$ minus the south pole and then $\varphi : U \to \mathbb R^2$ the stereographic projection $(X,Y)\mapsto ({x\over 1-z}, {y \over 1-z})$.
I have no trouble visualising these maps. But there are several things that bother me. 
For example, $S^2$ is two dimensional. This is supported but the fact that there are two component coordinates $(X,Y)$. But: in order to define these one assumes a point on $S^2$ is given by $(x,y,z)$. Cartesian coordinates in $\mathbb R^3$. I used to think that if a manifold has $n$-dimensions a point on it is given as an $n$-tuple $(x_1, x_2, \dots, x_n)$ which is used to define a map into $\mathbb R^n$.

Please could someone tell me what I misunderstand about coordinate
  charts?

The second thing that I am confused about is this:

When people write something like local coordinates $x_1, \dots, x_n$ for
  the manifold $M$ does one really mean charts $\varphi_1, \dots,
 \varphi_n$?

The coordinate charts define what each point $(x_1, \dots, x_n)$ is so I assume one cannot speak meaningfully of coordinates of a manifold unless one is talking about charts. Or am I missing something? 
When I am writing about coordinates I use $(x_1, \dots, x_n)$ to denote a bunch of numbers, like $(1,2,3,4)$, basically the coordinates of a vector as in linear algebra and coordinate charts are the maps that transport a similar bunch of numbers in $\mathbb R^4$ to $(1,2,3,4)$. 
I want to define a contact structure on a manifold. Some easy examples in  order to better understand. But:
The standard contact structure on $S^3$ is given by the contact form $xdy - y dx$. 

If we'd have the stereographic projection charts for $3$-dimensions,
  as above, would this contact form now be $XdY - Y dX$ or really just
  $xdy - y dx$?

 A: When you say you have no problems visualising the stereographic projections you (probably) see the sphere as embedded in $\Bbb R^3$. 
To define a chart you have to tell which point in $U$ goes where. You can't do this unless you give the points names a priori to having access to a chart. In the special case where the manifold in question lives inside some $\Bbb R^N$ ($N > n$) it might be convenient to use their Cartesian coordinates in $\Bbb R^N$ as such names. Once you defined a chart you can use its inverse to refer to the manifold points. Now you are left with an "$n$-dimensional namespace": an $n$-tuple is enough to reference a point. If a point $p \in \Bbb R^n$ references a manifold point $q$ via a chart, then there is a neighbourhood of $p$ in $\Bbb R^n$ which is diffeomorhic to a neighbourhood of $q$ in $M$. By contrast, in each neighbourhood of $q$ in $\Bbb R^N$, there are point, which are not used as names for manifold points. That's why $N$ doesn't say anything about the dimension of your manifold, whilst $n$ is actually used to define the dimension of $M$.
"When people write something like local coordinates $x_1,…,x_n$ for the manifold M", they usually refer to a single chart $x: U \to U' \subset \Bbb R^n$ and use $x_1, \dots x_n$ as abbreviatons to the projections of $x$ to it's component in $\Bbb R^n$, precisely: $x_i := pr_i \circ x$, where $pr_i: \Bbb R^n \to \Bbb R$ is the projection to the $i$th component relative to the standard coordinates of $\Bbb R^n$. So it is actually the chart combined with what you do, when you use coordinates in linear algebra.
Unfortunatly I can't help you with the part of your question concerning the contact structure. But as I understand the link you gave, you missed some termes in the definition of the standard contact strucure? Because $S^3$ lives in $\Bbb R^4$, so you can name it's points after their cartesean coordinates $(x_1, y_1, x_2, y_2)$ and then the contact structure would in these cartesean coordinates be given as $\alpha_0 = x_1 d y_1 - y_1 d x_1 + x_2 d y_2 - y_2 d x_2$.
