An algebraic structure is here taken in a traditional sense as a set with a (not necessarily finite) number of operations of various arity and satisfying some axioms (not necessarily of first order). That is, we are discussing objects of an equationally presentable or algebraic category.
An -group is an algebraic structure which amounts to a group (usually written additively but not necessarily commutative) together with a set of operations of any arity, such that for each -ary operation , the distributivity in each variable holds:
The classical examples are of course groups and rings, but also modules over a fixed ring: each element of the ground ring is a unary operation. An older term group with operators is traditionally used for -groups when only unary operations/operators are considered. The general theory of -groups is similar to the basics of group and ring theory, including ideals, quotient -groups, isomorphism theorems, etc. For example, the Jordan–Holder theorem holds: if there is a composition series, then every two composition series are equivalent up to permutation of factors. An obvious horizontal categorification of -groups is also interesting.
Wikipedia, Group with operators.
N. Bourbaki, Algebra I, ch. 1-3.
E. I. Khukhro, Local nilpotency in varieties of groups with operators, Russ. Acad. Sci. Sbornik Mat. 78 379, 1994. (doi)
Grace Orzech, Obstruction theory in algebraic categories I, II, J. Pure Appl. Algebra 2 (1972) 287-340, 315–340.