Let $k$ be an algebraicly closed field.

and let $\mathbb A^n(k)$ be our affine n space:

For $n=1$ we can clasify our closed zariski subsets of 1-space:

These are finite point sets and whole space and empty set.

For $n\ge 2$ what is the good approach to understand general closed subsets? I am revising my commutative algebra course and a bit confused and want to see these closed subsets with intuitive and good explanatory way.