An algebra is congruence modular (or CM for short) if its lattice of congruence relations is modular.

A class of algebras is congruence modular if each of its members is congruence modular.

Congruence modularity holds for many 'classical' varieties such as groups and rings.

A Mal'cev condition (with 4-ary terms) for congruence modularity is given by Alan Day, A characterization of modularity for congruence lattices of algebras., Canad. Math. Bull., 12, 1969, 167-173 MRreview

Another Mal'cev condition (with ternary terms) for congruence modularity is given by H.-Peter Gumm, Congruence modularity is permutability composed with distributivity, Arch. Math. (Basel), 36, 1981, 569-576 MRreview

Several further characterizations are given by Steven T. Tschantz, More conditions equivalent to congruence modularity, Universal algebra and lattice theory (Charleston, S.C., 1984), Lecture Notes in Math. 1149, 1985, 270-282, MRreview

Congruence n-permutable for $n=2$ or $n=3$.

Congruence distributive