How can I determine when two quotient rings of a polynomial ring are isomorphic? For example, is $F[x]/(x^2) \cong F[x] / (x^2 - x)$? I know (or at least I think) that they are isomorphic as additive groups, but I don't think they are as rings. How can I show this? In general, is there some criterion for determining when two quotient rings by ideals generated by polynomials of the same degree are isomorphic?
I doubt that a general criterion exists, but the Chinese remainder theorem can sometimes be helpful in simplifying quotient rings. In your example, the ideals $(x)$ and $(x-1)$ are coprime since $1=x-(x-1)$, hence $$ F[x]/(x^2-x)\simeq F[x]/(x)\times F[x]/(x-1)\simeq F\times F $$
If $F$ is a field then $F\times F$ has no nilpotent elements. On the other hand, $F[x]/(x^2)$ does have a nilpotent element, namely the image of $x$. So $F[x]/(x^2-x)$ and $F[x]/(x^2)$ are not isomorphic.