Zero locus vs Affine varieties?? [closed]

A subset $X \subset \mathbb{A}^n$ is called an affine variety if it is equal to $V(S)$ for some $S \subset K[x_1, \dots, x_n]$. For example, the set $\{(0,0)\} \subset \mathbb{A}^2$ is an affine variety because $$\{(0,0)\} = V(\{x_1, x_2\}).$$ $\{(0,0)\}$ is also equal to $V(\{x_1^2, x_2^2\})$, and that's fine -- as long as there is a set $S \subset K[x_1, \dots, x_n]$ such that $X = V(S)$, then we say that $X$ is an affine variety.

For a negative example, when $K = \mathbb{C}$, the set $\mathbb{N} \subset \mathbb{A}^1$ is not an affine variety. To see this, suppose for contradiction that $V(S) = \mathbb{N}$ for some $S \subset \mathbb{C}[x]$. Then $S$ is nonempty since $V(\varnothing) = \mathbb{C} \neq \mathbb{N}$, so pick some $f \in S$. Now $f \in \mathbb{C}[x]$ has finitely many roots, so $V(S) \subset V(\{f\})$ is finite. This contradicts the fact that $\mathbb{N}$ is infinite.

Locus is a more general term, meaning more or less a set points satisfiying "something". Zero locus then means the set of points where "some function(s)" vanish(es).

In your source, an affine variety is then defined as the zero locus of a collection of polynomials over an affine space, i.e. the set of points in an affine space $\Bbb A^n$ where every polynomial in a collection of polynomials vanishes simultaneously.