Such a ring has simple socle, namely $\mathfrak m^n$. It follows easily that $A$ is self-injective, Gorenstein and, of course, of dimension $0$. You can construct them using Macaulay's method of inverse system; this is explained in Eisenbud's book on commutative algebra, if I recall correctly.
If $A_4=\mathbb C[x,y]/(x^2,y^2)$, a four-dimensional example, then you cannot embed it in an algebra of the form $\mathbb C[t]/(t^\ell)$.
Suppose now $A$ is an algebra satisfying your conditions with $n\geq3$, and let $x$, $y\in\mathfrak m^{n-1}\setminus\mathfrak m^{n}$. Then $xy\in\mathfrak m^{2n-2}\subseteq\mathfrak m^{n+1}=0$. If $\dim\mathfrak m^{n-1}/\mathfrak m^{n}\geq2$, we then see that $A$ contains a subalgebra isomorphic to $A_4$, so $A$ itself cannot be embedded in an algebra of the form $\mathbb C[t]/(t^\ell)$. It follows that if $A$ does embed and $n\geq3$, then $A/\mathfrak m^n$ satisfies the same conditions as $A$ with an $n$ smaller by $1$. From this one should be able to describe which algebras embed.