Let $S^1 = \mathbb{R}/\mathbb{Z}$ and for $\alpha \in \mathbb{R}/\mathbb{Q}$ define the rotation $$R_\alpha(x) = x + \alpha \mod 1.$$ I don't understand who is $$R_\alpha(1) - \alpha.$$ Is $R_\alpha(1) - \alpha = (1 + \alpha \mod 1) - \alpha = \{1 + \alpha\} - \alpha$, where $(1 + \alpha \mod 1) = \{1 + \alpha\}$ is the fractional part of $1 + \alpha$?
P.S. Can someone explain me who is $R_\alpha(x) - \alpha$ in general?
Everything is in the definitions. First, a point on the circle $S^1$ is parametrized by an angle $x\in \mathbf R$, but this parametrization is not a bijection, $x$ being defined only up to an integer multiple of say $2\pi$. So that actually $S^1\cong \mathbf R/2\pi \mathbf Z\cong \mathbf R/\mathbf Z$ as additive groups; this is indeed the correct mathematical definition of angles and their addition law, the last isomorphism amounting to the choice of a unity of measurement). Second, how to define what you call the group, say $Rot$, of rotations $R_\alpha$ ? If we do not put any restriction on $\alpha$, the previous definition of angles would imply $Rot\cong S^1$. So your written isomorphism $Rot\cong \mathbf Q/\mathbf Z$ means that you actually consider only rotations whose angles are rational multiples of $2\pi$.
This being settled, let us slightly abuse language by writing also $x$ and $\alpha$ for representatives of resp. $x\in S^1$ and $\alpha \in Rot$. Then, by the addition law defined above, $R_\alpha (x)\in S^1$ is represented by the angle $x+\alpha+2k\pi$, and $R_\alpha (x)-x$ by $2k\pi$. Geometrically, if you choose (necessarily in an arbitrary way) a system of coordinates in the plane, the point $R_\alpha (x)-x$ could be viewed as the intersection $A$ of the circle and the abscissa axis. What about $R_\alpha (1)$ ? Your notation $1\in S^1$, given our previous definitions, should mean the point of the circle defined by the angle = say $1$ radian. But I guess that there could be a confusion in your mind with the intersection point $A$ just introduced.