Support of a module $M$ is  $$   \text{supp}(M) = \{\mathfrak{p} \in Spec R: M_\mathfrak{p} \neq 0\} $$ which is basically the collection of points in the $SpecR$ that is non-trivial under the localization of the module to the point $\mathfrak{p}$.

Now, we move on to defining the associated primes of the module. There are finitely many associated points in $SpecR$.

An associated point is called an embedding point if it is in the closure of some other associated point. This means that if $R$ can be reduced, then there are no embedded points.

We also define that $$ \text{Supp}(M)= \overline{Ass_R M} $$ where $M$ is an $R$-module. This means that the support of $M$ is the closure of the associated points of $M$.

We call an element of $R$ a zero-divisor if it vanishes at an associated point. Now we define annihilator ideal $Ann_R\ m$ for $m \subset R$ of a module $M$ $$Ann_R\ m := \{a \in R: am =0\}.$$

There is the localization at these associated primes. Given that $R$ is a Noetherian ring and a module M over $R$, then there is an injection $$ M \rightarrow \Pi_{\mathfrak{p}\in Ass M} M_\mathfrak{p}$$ with a similar injection in the localization at the $SpecR$. (Support of the module and the associated primes commute with localization.) For any reference, see the frequently mentioned notes on this blog.