It is the mathematical logic counterpart of the notion in category theory of (the codomain of) a forgetful functor, and in physics of (hypothesized) hidden variable theories purporting to explain quantum mechanics.
That is, it is obtained by omitting some of the sorts and relations of a (many-sorted) elementary class.
A quasivariety defined logically as the class of models of a universal Horn theory can equivalently be defined algebraically as a class of structures closed under isomorphisms, subalgebras, and reduced products.
A corollary of this characterization is that one can (nonconstructively) prove the existence of a universal Horn axiomatization of a class by first axiomatizing some expansion of the structure with auxiliary sorts and relations and then showing that the pseudoelementary class obtained by dropping the auxiliary constructs is closed under subalgebras and direct products.
This short proof is an effective application of abstract nonsense; the stronger result by Tarski that RRA is in fact a variety required more honest toil.