Let $k$ be algebraically closed of characteristic zero; assuming $k=\mathbb{C}$ is fine. Consider the subring inclusion $k[z^2]\to k[z]$ and the induced map $$ f\colon\mathbb{A}^1_k=\mathrm{Spec}k[z]\to\mathbb{A}^1_k=\mathrm{Spec}k[z^2] $$ given by $f(\mathfrak{p})=\mathfrak{p}\cap k[z^2]$. This map satisfies $f((z))=(z^2)$ and more generally $$ f((z-a))=(z^2-a^2)=f((z+a)). $$
What is the name for the relationship between these two affine lines and for whatever this phenomenon generalizes to? It seems like we have a twofold cover that ramifies at the origin, but aren't those usually induced by power maps $z\mapsto z^n$? It seems like we should have a section of the cover induced by the power map, but $f$ is two-to-one and didn't involve a choice of square root, so this doesn't seem correct.
Yes, this is a branched cover, unramified away from $0$, and is indeed induced by the map $z \mapsto z^2$. In fact you have observed that the maximal ideals $(z \pm a)$, which correspond to $\mp a \in k$ are mapped to the maximal ideal $(z^2 - a^2)$, corresponding to $a^2$.