What is the difference between "a free group generated by A" and "a group generated by A"?
I saw the definition, but I don't know. However, since $\mathbb{Z}\times \mathbb{Z}$ is generated by two elements and is abelian, it is not a free group, so I understand that the two definitions are different.
Let me add a few examples. If $A=\{a\}$, then the free group generated by $A$ has the presentation $\langle a\rangle$ and is given by $$ \langle a \rangle =\{a^k\mid k\in \Bbb Z\}\cong \Bbb Z, $$ whereas the group generated by $A$ also can have a relation $a^n=1$ for some $n$, so that the group is given by $$ \langle a \mid a^n=1\rangle \cong C_n, $$ which is a finite cyclic group.
For $A=\{a,b\}$, the free group generated by $A$ is $$ F_2=\langle a,b\rangle. $$ This is an infinite, non-solvable group, which is quite different from "the free abelian group" $$ \langle a,b\mid ab=ba\rangle \cong \Bbb Z\times \Bbb Z. $$ Examples of special properties of $F_2$ include the fact that $F_2$ contains every free subgroup $F_k$ of rank $k\ge 2$.
The free group $F_2$ contains $F_k$