Let $R$ be a commutative unitary ring that need not be local. Let $N$ be a nonzero $R$-module.
Suppose there exists a nonzero element $x$ of $N$ and a maximal ideal $\mathfrak{m}$ of $R$ such that $x$ is killed by a power of $\mathfrak{m}$. Let $I$ be ann$(x)$. Then $Rx$ is isomorphic to $R/I$. By the assumption, there exists an integer $n > 0$ such that $\mathfrak{m}^n \subset I$. We can assume that $n$ is the least such integer. There exists $a \in \mathfrak{m}^{n-1} - I$. Since $\mathfrak{m}a \subset \mathfrak{m}^n \subset I$, ann($a$ mod $I) = \mathfrak{m}$. Hence $R/I$ contains a submodule isomorphic to $R/\mathfrak{m}$. Hence $Rx$ contains a simple submodule.
Conversely suppose $N$ has a simple submodule $L$. $L$ is of the form $Rx$, where $x$ is a nonzero element of $N$. Let $\mathfrak{m}$ be ann($x$). Since $L$ is isomorphic to $R/\mathfrak{m}$, $\mathfrak{m}$ is maximal. Clearly $\mathfrak{m}x = 0$.