Help with notation in second order tensor. I have seen recently this notation:
$$F=F_{ij}\,e_i\otimes e_j $$
Where $F_{ij}=\frac{\partial x_i}{\partial X_j}$ is the tensor matrix:
$$      F=\left[ {\begin{array}{cc}
             \frac{\partial x_1}{\partial X_1} & \frac{\partial x_1}{\partial X_2} & \frac{\partial x_1}{\partial X_3} \\
 \frac{\partial x_2}{\partial X_1} & \frac{\partial x_2}{\partial X_2} & \frac{\partial x_2}{\partial X_3} \\
 \frac{\partial x_3}{\partial X_1} & \frac{\partial x_3}{\partial X_3} & \frac{\partial x_1}{\partial X_3} \\
                \end{array} } \right]
$$
I don't understand the intuition behind that notation neither what is useful for.

The formula you write refers, in general, to an element in the tensor product of modules over a ring, or vector spaces over a field. 
Let us try to write an example involving the explicit definition of $F_{ij}$.
Let $x_i:=f_i$, with $f_i=f_i(X_1,X_2,X_3)$ denote the components of a $C^1$ function $f:\mathbb R^3\rightarrow \mathbb R^3$. The matrix $F$ with entries $F_{ij}$ is just the Jacobian of $f$ at $(X_1,X_2,X_3)$ w.r.t. the basis $\{e_1,e_2,e_3\}$ of $\mathbb R^3$ (usually one uses the canonical basis). With some abuse of notation we write $F\in\operatorname{End}_{\mathbb R}(\mathbb R^3)$.
Let us denote by $\{\epsilon_1,\epsilon_2,\epsilon_3\}$ the dual basis on ${(\mathbb R^3)}^{*}$. Using the chain of isomorphisms of vector spaces over $\mathbb R$
$$\operatorname{End}_{\mathbb R}(\mathbb R^3)\stackrel{I}{\leftarrow} \mathbb R^3\otimes {(\mathbb R^3)}^{*}\leftarrow \mathbb R^3\otimes\mathbb R^3,$$
where the leftmost isomorphism is given by
$$I(e_i\otimes \epsilon_j)(e_k):=e_i\epsilon_j(e_k)= e_i\delta_{jk},$$
one can identify $F\in\operatorname{End}_{\mathbb R}(\mathbb R^3)$ with the element in $\mathbb R^3\otimes\mathbb R^3$ given by $F_{ij}~e_i\otimes e_j$ (using Einstein notation).
