Algebraic structure may mean either There are several notions of
an algebraic structure on an object of some category or higher
category, which differ in generality. It may be to be an algebra
over an algebraic theory, algebra over an operad (or higher
operad) or an algebra over a monad, or over a prop, over a
properad etc. See also variety of algebras. There is also an
older notion of an algebraic structure/algebra as a model for a
one-sorted theory where the only relation symbols in the
language involved are and equality (with standard interpretation
in models). This notion includes for example fields which are
not an algebraic theory in the sense of monads (because there
are no free objects in the case of fields, i.e. the category of
fields is not monadic over the category of sets). There is a
forgetful functor from the category of algebras/algebraic
structures of some type (in any of the above formalisms) to the
original category. This functor forgets structure in the sense
of stuff, structure, property.