In Commutative Algebra, we defined the Support of a Module $M$ $$ \operatorname{Supp}(M) = \{P \in \operatorname{Spec}(R) : M_P \neq \{0 \} \} = \{P \in \operatorname{Spec}(R): \exists m \in M: \operatorname{Ann}(m) \subseteq P \} $$ Today, I asked our tutor "Can you tell us why $\operatorname{Supp}(M)$ is interesting?" and he told me no, that he has never really seen it in action.
So I want to pose this question here. Why should $\operatorname{Supp}(M)$ be interesting?
Consider a sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules. Then the support of $\mathcal{F}$ is all the points of $X$ "where the stalk is non-zero", i.e. $$\mathrm{supp}(\mathcal{F}) = \{ x \in X \mid \mathcal{F}_x \neq 0 \}. $$ This indeed coincides with the definition you've given in your question. Note that one can generalise this notion to complexes of sheaves, i.e. $$ \mathrm{supp}(\mathcal{F}^\bullet) = \bigcup \mathrm{supp}(H^i(\mathcal{F}^\bullet)). $$ Then this notion is important in stating/proving many results from algebraic geometry. For example, if we know somthing like $x \notin \mathrm{supp}(\mathcal{F}^\bullet) $ then we can deduce that $\mathcal{F}^\bullet |_U$ is trivial, where $x \in U \subset X$ is an open neighbourhood of $x$.