Let $R$ be a commutative ring with 1 such that it is local with a unique non-idempotent minimal ideal, that is, there are two ideals $I$ and $m$ of $R$ such that for each ideal $K$ of $R$ with $0\not=K\not=R$ we have $I\subseteq K\subseteq m$ and $I^2=0$. I am looking for a characterization for such a ring.
It seems to me that if $R$ is Noetherian, then this implies that it is an Artinian Gorenstein ring and conversely the condition holds for those. Here is why:
By your condition $I\subseteq \mathfrak m^n$ for every $n$ for which $\mathfrak m^n\neq 0$, but by the Krull intersection theorem the intersection of all of those is zero which implies that there must be an $n$ for which $\mathfrak m^n= 0$. This implies that $R$ is Artinian.
Your condition also implies that then $I$ is the socle of $R$ and it has to have dimension $1$ as a vector space over the residue field of $R$. That implies that $R$ is Gorenstein.
Conversely, the socle of an Artinian Gorenstein ring has dimension $1$ as a vector space over the residue field of $R$ and it is an essential submodule of $R$ which implies both of your conditions.