Some questions about the cartesian product I understand that the cartesian product of $A \times B$ is a set with elements of the form $(a,b)$ where $a\in A$, $b\in B$.
My question arise from the fact that I was described $\Bbb{R}^3$ as $\Bbb{R} \times \Bbb{R} \times \Bbb{R}$, but elements of $\Bbb{R}^3$ have the form $(x,y,z)$, while elements of $\Bbb{R} \times \Bbb{R} \times \Bbb{R}$ should have the form $((x,y),z)$ where $(x,y)\in \Bbb{R}^2,z\in\Bbb{R}$. 
If this sets are different, how do we construct $\Bbb{R}^n$ with elements of the form $(x_1,x_2,...,x_n)$?

$A \times A \times A$ is usually defined as $(A \times A) \times A$ when the Cartesian product of two sets has been defined. This corresponds to your first view of $\mathbb{R}^3$.
On the other hand, powers of sets are also defined, namely $A^B$ is defined as the set of all functions from $B$ to $A$. Defining the natural number $n+1$ as the set $\{0,\ldots,n\}$ (and $0 = \emptyset$), as is usual, we can define $A^n$ as the set of all functions from the set $n$ to $A$.
It's quite easy to see that we can identify an $f \in A^2 = A^{\{0,1\}}$ with its tuple of values $(f(0), f(1))$ and so with $A \times A$ as a Cartesian product, and similarly $A^3$ with $(A \times A) \times A$ etc. so that (up to obvious bijections; the sets are not the same as pure sets, but can be easily identified using "trivial"  or "natural" bijections) we can consider powers of a set as iterated products (like we have for numbers). The view of $\mathbb{R}^n$ as $n$-tuples corresponds to the "power" view most naturally, but as said, is easily identified with iterated Cartesian products as well.
Also see this answer, e.g.
