Explanation of Set Notation In a set, for example the following one

$$\{{(x_{1},x_{2}) \in \mathbb{R}^{2}}|x_{1}+x_{2}=1\}$$

does the notation $(x_{1},x_{2}) \in \mathbb{R}^{2}$ means that $x_{1}$ and $x_{2}$ are points in $\mathbb{R}^{2}$ space or it means that they are the coordinates of the points in that space?

For example if I have two points $p_{1}=(px_{1},py_{1})$ and $p_{2}=(px_{2},py_{2})$ in $\mathbb{R}^{2}$ does it mean that

$$px_{1}+py_{1}=1$$

or that

$$p_{1}+p_{2}=1$$
The set $$\Big\{ (x_1,x_2)\in\mathbb{R}^2\mid x_1+x_2=1\Big\}$$ 
 considers all points in $\mathbb{R}^2$ (that is, ordered pairs with real coordinates) such that they obey a property (in this case, the sum of their coordinates is $1$).

For example, point $\left(\dfrac12, \dfrac12\right)$ belongs to such set, because $\dfrac12 + \dfrac12 = 1$. 

If you consider the set of points in $ \mathbb{R}^2 $ that do not obey this property, you can write
$$ \Big\{ (x_1,x_2)\in\mathbb{R}^2\mid x_1+x_2\neq1\Big\}.$$
Set builder notation, like what you are asking about here, is usually in the form of $\{x\in S | \phi(x)\}$, where the elements of the set are of the form $x$, they belong to the ambient set $S$, and they all satisfy the formula $\phi(x)$.

Here, our set reads $\{(x_1,x_2)\in\mathbb{R}^2 | x_1+x_2=1\}$. Our elements are ordered pairs, belonging to $\mathbb{R}^2$, with the property that their sum of the coordinates is $1$.