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. We say that a functor in the base category preserves some algebraic structure if it lifts to the corresponding category of algebras.