What does $\overline{r}m:=rm$ mean? On this Wikipedia article, it says that you can define an $R$-module $M$ as an $R/Ann_R(M)$-module using the action $\overline{r}m:=rm.$ 
What does that action actually mean? What is $\overline{r}$?

When a projection map $R\to S$ is in play, the notation $\bar{r}$ often denotes the image of $r$ in $S$ under the projection map. E.g. the elements of $\Bbb Z/n\Bbb Z$ are the residues $\overline{0},\overline{1},\overline{2},\cdots,\overline{n-1}$.
A priori one would have to wonder if $\overline{r}m:=rm$ is well-defined, since $\overline{r}=\overline{s}$ is possible in the quotient ring $R/{\rm Ann}_R(M)$ even if $r\ne s$ in $R$, and a priori different elements of $R$ act on the elements $m\in M$ in different ways. But if $r-s\in{\rm Ann}_R(M)$ then $(r-s)m=0$ implies that the elements $rm=sm$ are equal, and so $\overline{r}m$ is independent of choice of representative $r\in R$ for the residue $\overline{r}\in R/{\rm Ann}_R(M)$.
