I'm studying algebraic geometry from the classical viewpoint in which the Zariski topology takes center stage and schemes have yet to be invented. I sometimes see the term "closed subvariety" thrown around, but I can't find a proper definition for this. For example on p.43 of J.S. Milne's notes we find:
PROPOSITION 2.27. Let $V$ be an irreducible variety such that $k[V]$ is a unique factorization domain. If $W\subseteq V$ is a closed subvariety of dimension $\mathrm{dim}(V)-1$, then $I(W)$ is a principal ideal.
(In this context $I(W)$ means the set of all polynomials that vanish on $W$.)
I'm not quite sure what a closed subvariety is. Isn't every variety automatically closed?
Maybe your confusion lies in the difference between algebraic sets, affine algebraic varieties and algebraic varieties. By algebraic sets I mean the zero sets of polynomials whereas (affine) varieties are defined as ringed spaces. But depending on the author these can mean different things.
It seems like your are using an old version of the notes by Milne. I would recommend you to switch to the newest version (6.02) of the notes. There he starts by defining algebraic sets and later moves to (affine) algebraic varieties. On page 68 is a definition of closed subvarities.