Can you give an interesting example of useage of this Categorical definition of free object? 
Definition 1.2 If $C$ is a concrete category, $F \in Obj(C)$, and $X$ is a set,
  and if $\phi : X \to \sigma(F)$  is a one-to-one function, then $F$ is called free on $X$ if and only if for every $A \in Obj(C)$ and set map $\psi : X \to \sigma(A)$, there exists a unique morphism $f \in Mor(F, A)$ such that $f \circ \phi = \psi$ as set maps from $X$ to $\sigma(A)$. 

$\sigma$ is the map that comes together with the concrete category $C$, the map that makes $C$ into a concrete category.
Thanks.  It's just that this definition is so abstract. 
 A: Consider the category $\mathbf{\mathbb{K}\text{-}Vect}$ of $\mathbb{K}$-vector spaces. Let $V$ be a vector space, $B$ its basis and $\phi\colon B\to\sigma(V)$ the obvious embedding. Then $V$ is free on $B$, because the condition for freeness translates exactly to the property, that any linear map between vector spaces is completely determined by its action on the basis vectors.
A: A way to think about it : a free object on the set $X$ respects the universal property that $F(X)$ would respect if the forgetful functor $U \colon C \to \mathbf{Sets}$ had a left adjoint $F$.
So now, just think about what is a left adjoint of $U$ : it is a functor $F \colon \mathbf{Sets} \to C$ which comes with a (natural) bijection for all object $Y$ of $C$
$$ \hom_{C}(F(X), Y) \simeq \hom_{\mathbf{Sets}}(X,U(Y)).$$
It means that defining a map from a free object on the set $X$ to the object $Y$ is the same as defining a map form $X$ to the underlying set of $Y$. Which is precisely what you do with the usual free object you (should) know : free groups, free modules, etc.
A: You shouldn't learn category theory now if you don't know immediately at least three examples for free constructions. Or you should choose another textbook, because any good textbook mentions basic examples from which one then abstracts.
