According to Wikipedia:
Let G be a group, written multiplicatively, and let $R$ be a ring. The group ring of $G$ over $R$, which we will denote by $R[G]$ (or simply $RG$), is the set of mappings $f : G \to R$ of finite support, where the module scalar product $\alpha f$ of a scalar $\alpha$ in $R$ and a vector (or mapping) $f$ is defined as the vector $x\mapsto \alpha \cdot f(x)$.
My current understanding is that the set $R[G]$ contains all the possible mappings $f : G \to R$ with the following property: $f(x)= 0$ for all but finitely many $x\in G$. I'm not sure how to interpret "the module scalar product $\alpha f$ of a scalar $\alpha$ in $R$ and a vector (or mapping) $f$ ". Moreover, what is the domain of these mappings? The group $G$ itself?
The product of two mappings $f$ and $g$ is defined as: $x\mapsto \sum\limits_{u\in G}{f\left( u \right)g\left( {{u}^{-1}}x \right)}=\sum\limits_{uv=x}{f\left( u \right)g\left( v \right)}$. I understand that, for instance, $f\left( u \right)\ne 0$ only for a finite number of $u$'s, therefore this definition is legitimate. What I don't understand is why those two definitions are equal.
This is just an intricate way to say that $\alpha\cdot f$ is the function $\alpha f$ (defined in the last part of your sentence).
Yes.
Well, just set $v=u^{-1}x$ (if if $uv=x$, then $v=u^{-1}x$ ande vice versa).