Groups, semigroups, fields, rings, integral domains, vector spaces, R-modules... these are all approximately the same sort of "stuff", but each one refers to a slightly different combination of required properties. Is there a general term that collectively refers to these types of "stuff"? Also, which branch of mathematics do such objects live in (broadly speaking)?
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All of the objects you listed could be collectively called "algebraic objects" and their theories individually are usually considered to be under the purview of abstract algebra. But of course nearly every other field uses them too!
The study of algebraic objects in the abstract is called universal algebra. The generic name for algebraic objects in universal algebra is that they are all "algebras" of different types. (This is not to be confused with the other very widespread term for an associative ring with a scalar operation over a field.)
They go so far as to define $n$-ary operations on sets. Many different theorems (for example, the basic isomorphism theorems) which are usually proven separately for all those objects are proven all at once for large classes of algebraic objects in the eyes of universal algebra.
They fit under the umbrella of Universal Algebra, which collects & studies algebras, which are mathematical artifacts with operations only. See this excellent book for details.