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 $A=\mathbb C[x,y]/(x^2,y^2)$, a four-dimensional example, then you cannot embed it in an algebra of the form $\mathbb C[t]/(t^\ell)$.