For some varieties $V$, for any algebra $\mathbb{A}\in V$, there exists a lattice isomorphism $f:\text{Con}(\mathbb{A})\to \mathbb{B}$ where $\mathbb{B}\leq \text{Sub}(\mathbb{A})$ between the lattice of congruences of $\mathbb{A}$ and a sublattice of subalgebras of $\mathbb{A}$, and the choice of $\mathbb{B}$ is natural in some sense.
For instance, if $V$ is the variety of groups, any congruence corresponds to a normal subgroup, and if $V$ is the variety of rings, congruence corresponds to an ideal.
Can someone point me to literature where such varieties are studied? Names, anything? Proper definition? It doesn't have to be precisely what I've written.
I think that you are looking for the phrase ``ideal determined variety''. Might want to start with this paper:
http://www.mathematik.uni-marburg.de/~gumm/Papers/Ideals%20in%20universal%20algebras.pdf