Right translation - left coset - orbits We can remark that the left coset $gH$ of $g \in G$ relative to a subgroup $H$ of $G$ is the orbit of $g$ under the action of $H \subset G$ acting by right translation. 
What is that right translation? and how  can I prove that the orbit of $g$ under the action of $H \subset G$ acting by right translation is $gH$ ? 

Right translation can equally be read as "right multiplication", except there is an implication of commutativity. 
As to your second query, let the subgroup $H$ act on $G$ by right multiplication:
$$h \cdot g = gh \qquad \forall g \in G \quad \forall h \in H$$
For any $g \in G$, the orbit $H \cdot g$ is the set
$$\{ h \cdot g : h \in H \} = \{ gh: h \in H \} = gH.$$
Note that we didn't need the operation to be commutative here.
