Given a left action $(g,x) \to g.x$ of $G$ on $X$, you can define a corresponding right action by defining $(x,g) \to g^{-1}.x$.
The "$a$" in the definition is any element of $G$: .
So the left coset $\,aH\subseteq G\,$ is the set of all elements in the left coset $aH$, which for a given $\,a \in G\,$ and every element $h_i \in H$, is the set of all $ah_i$.
E.g. Take a small subgroup of $S_3$ : $\;H = \langle (12)\rangle = \{id, (12)\} \leq S_3.\,$ There are three left (respectively right) cosets of $\,H$ in $\,S_3$. One coset is $\,H\,$ itself. The other cosets are $\,(13)H = (123)H\,$ and $\,(23)H = (132)H$.
You'll see that for any subgroup $\,H \leq G$, every element of $\,G\,$ will belong to one and only one left (respectively right) coset of $\,H\,$ in $\,G.\,$ And the union of all left cosets of $H$ in $G$ (respectively the union of all right cosets of $H$ in $G$) is $G$. That is, the left (respectively right) cosets of $H$ in $G$ partition $G$.
You'll can find a nice definition of "coset" and some examples here, as well.
Best Answer
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.