What is the definition of a free abelian group generated by the set $X$? What is the definition of a free abelian group generated by the set $X$?
If $X=\{A,B,C,D\}$ what are the elements of the free abelian group on this set?  What is the identity of this group?
 A: It is the abelian group $ \mathbf Z^{(X)}$ i.e. the set of functions $f:X\rightarrow \mathbf Z$ with finite support. If, as is usual, we represent such a function  by its values at each point $x\in X$, it is
$$ \mathbf Z^{(X)}=\Bigl\{(n_x)_{x\in X}\mid n_x=0\enspace\text{except for a finite number of them}\Bigr\}.$$
If $|X|=n$, it'is the same as $\mathbf Z^X=\displaystyle\prod_{x\in X}\mathbf Z$ and it's  isomorphic to $\mathbf Z^n$.
Note: as an abelian group, the ring of polynomials with integer coefficients is defined  as $\mathbf Z^{(\mathbf N)}$, to which one adds an extra multiplicative structure.  In contrast, $\mathbf Z^{\mathbf N}$ with the extra multiplicative structure given by the Cauchy product, is the ring of formal power series with integer coefficients.
A: One way you can think of this in a very elementary and canonical way is that the free abelian group on a finite set such as $\{A,B,C,D\}$ is the group of all functions
$$f : \{A,B,C,D\} \to \mathbb{Z}
$$
with the operations of pointwise addition and pointwise inverse: given any two such functions, for all $x \in \{A,B,C,D\}$ define
$$(f_1+f_2)(x) = f_1(x) + f_2(x)
$$
also
$$(-f)(x) = -f(x)
$$
The identity element, of course, is the constant function $f(x)=0$.
For infinite sets, you have to add one more clause to the definition: for the free abelian group on a set $X$, you do not take all functions $f : X \to \mathbb{Z}$, you take only those functions which satisfy the property that there is a finite subset $Y \subset X$ such that if $x \in X-Y$ then $f(x)=0$.
A: I preassume that here the elements $A,B,C,D$ are distinct. 
An abelian group free on set $\{A,B,C,D\}$ is the group $\langle\mathbb Z^4,+\rangle$ where $+:\mathbb Z^4\times\mathbb Z^4\to\mathbb Z^4$ is prescribed by:$$+(\langle n_A,n_B,n_C,n_D\rangle,\langle m_A,m_B,m_C,m_D\rangle)=\langle n_A+m_A,n_B+m_B,n_C+m_C,n_D+m_D\rangle$$ Or more commonly:$$\langle n_A,n_B,n_C,n_D\rangle+\langle m_A,m_B,m_C,m_D\rangle=\langle n_A+m_A,n_B+m_B,n_C+m_C,n_D+m_D\rangle$$
If $\eta:\{A,B,C,D\}\to\mathbb Z^4$ is prescribed by: 


*

*$A\mapsto\langle1,0,0,0\rangle$

*$B\mapsto\langle0,1,0,0\rangle$

*$C\mapsto\langle0,0,1,0\rangle$

*$D\mapsto\langle0,0,0,1\rangle$


Then for every function $f:\{A,B,C,D\}\to A$ where $A$ denotes an abelian group there is a unique groupmorphism $\phi:\langle\mathbb Z^4,+\rangle\to A$ such that $f=\phi\circ\eta$. 
This universal property is characteristic for a "free abelian group on a set".
More explicitly: $$\phi(n_A,n_B,n_C,n_D)=n_Af(A)+n_Bf(B)+n_Cf(C)+n_Df(D)$$
Here e.g. $3a$ where $a\in A$ must be looked at as a notation for $a+a+a$.
A: The free abelian group $G(X)$ generated by $X = \{A,B,C,D\}$ is the smallest abelian group containing $X$ and has no other restricting properties. You 
should think of $X$ as a basis for the group $G(X)$, similarly to a basis for a vector space.
Concretely, what are the elements of $G(X)$? Starting from $X$, first adjoin symbols that will represent the identity in $G(X)$ and inverses
for the generating elements: $0, −A, -B, -C, -D$. Now form the set of all abstract finite sums, e.g. $A + B + D + (-B)$. Finally, introduce relations (i.e. define the set of equivalence classes of sums) saying that these elements and sums behave like a commutative group; for example,
$A + B + D + (-B) = A + D$.
Since the group is abelian, we can collect terms; for instance, $A + B + A = A + A + B = 2A + B$. So, each element of
the free abelian group $G(X)$ can be expressed uniquely as a sum
$$
n_AA + n_BB + n_CC + n_DD 
$$
where each $n_x \in \mathbb{Z}$; in other words, $G(X) = \big\{\sum_{x \in X} n_X X \mid n_x \in \mathbb{Z} \big\}$. Therefore, you can easily see that $G(X) \simeq \mathbb{Z} \times \mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}$, where the group operation is the pointwise addition and the neutral element is $(0,0,0,0)$. 
In the explanation above, note that actually the nature of the elements of $X$  plays no role, the only relevant information is the number of elements of $X$. Thus, more generally, you can see that given a set $X$ with exactly $n \in \mathbb{N}$ elements, $G(X) \simeq \underbrace{\mathbb{Z} \times \dots \times \mathbb{Z}}_{n \text{ times } \mathbb{Z}}$.
A: Long story short: it is the set of all words in the four letters $A, B, C, D$ and their formal inverses $A^{-1}, B^{-1}, C^{-1}, D^{-1}$, where any two letters commute.
E.g. $ABBADC^{-1}$ is one such word, and by commutation it is equivalent to $AABBC^{-1}D$. It is not hard to see why this group is isomorphic to $\mathbb{Z}^4$ (via the powers in the notation $A^2B^2C^{-1}D^1$).
"Free" in this context is here to emphasise that there are no other relations than the commutation.
A: Other answers have already pointed out concrete models for free Abelian groups. I want to emphasize that they can be characterized by a universal property, which I found helpful to gain some perspective on the construction.
In order to state the universal property, note that every Abelian group $G$ has an underlying set $\underline G$. The passage from $G$ to $\underline G$ simply forgets the group structure. Similarly, a group homomorphism $\alpha\colon G\to H$ has an underlying map of sets $\underline \alpha\colon \underline G\to\underline H$.
Now let $S$ be a set. An Abelian group $F(S)$ together with a map $i\colon S\to \underline{F(S)}$ is called freely generated by $S$ if for every Abelian group $G$ and every map $a\colon S\to \underline G$ there is a unique group homomorphism $\alpha\colon F(S)\to G$ such that $a=\underline \alpha\circ i$. In other words, the map $i$ gives a bijection $$Hom(F(S),G) \cong map(S,\underline G),\quad \alpha\mapsto \underline \alpha\circ i.$$ This property characterizes the pair $(F(S),i)$ uniquely up to unique isomorphisms. It manifests the slogan that "homomorphisms out of $F(S)$ are determined by the values on $S$".
EDIT: By the way, if you erase the word "Abelian" everywhere, you get the universal property of free groups. In fact, I initially missed that word and wrote the answer for groups. Even more generally, you can speak of "free objects" in other categories.
