A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(A)\rightarrow X$ where $A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ is an infinitesimal thickening of $Y$, then $X$ is non-singular.

My question is:

if $k$ is algebraically closed, can we say explicitly what every local Artin ring finite over $k$ is?

$A$ can be of the form $k[t]/(t^n)$ or for instance if its maximal ideal isn't principally generated $k[t^2,t^3]/(t^4)$. Are there $A$ which we cannot write in this "adjoin various powers of $t$ and mod out by some power of $t$" form? If there are more exotic $A$ can we say anything non-tautological about the structure of such an $A$?

Thanks.