Concretely, every set function $f:A\rightarrow B$ induces an adjunction $f_\ast \dashv f^\ast$ between image and preimage. For rings groups, the image and preimage along a surjective ring group homomorphism preserve ideals normal subgroups, which are equalizers of arrows with the arrow that picks out $0$. This is no surprise for $f^\ast$ since it preserves equalizers, but not generally true for $f_\ast$.
Since rings groups are an algebraic theory, the regular epis are precisely the surjective functions. It seems one could just say that regular epis induce image-preimage adjunctions that preserve regular monos.
As pointed out in the comments, the case of rings is not as pleasant because ideals leave the category of rings, so maybe I should replace 'regular' by 'normal'...
At any rate, in which algebraic categories (with zero object) does this hold?
Firstly the comments are perhaps misplaced regarding rings since it depends whether or not the definition you use requires a multiplicative identity or not.
For normal epimorphisms and normal monomorphisms this condition has been studied and such categories are called Ideal-determined categories. Any variety of universal algebras that has a unique constant $e$ and for some natural number $n$ one $n+1$-ary term $p(x_1,..,x_n,x_{n+1})$ and $n$ binary terms $s_i(x,y)$ such that $s_i(x,x)=e$ and $p(s_1(x,y),..,s_n(x,y),y)=x$ satisfies this condition. These include the categories of groups, Lie algebras over a field (or even a commuative ring), not necessarily unital rings and many others. It is perhaps worth noting that there are examples that satisfy this condition that do not have such terms.