1) Characteristic $0$ is irrelevant: if $K$ is an algebraically closed field of any characteristic the maximal ideals of $K[x_1,\ldots,x_n]$ are the ideals $I_a=(x_1-a_1,...,x_n-a_n)$ consisting of the polynomials $f\in K[x_1,\ldots,x_n]$ vanishing on $a=(a_1,\ldots,a_n)\in K^n$.
2) For an arbitrary, not necessarily algebraically closed, field $K$ the maximal ideals of $K[x_1,\ldots,x_n]$ correspond to the closed points of affine space $\mathbb A^n_K$.
A rough description of these points is as follows:
Choose an algebraic closure $K^a$ of $K$. There is a canonical morphism of schemes $\mathbb A^n_{K^a} \to \mathbb A^n_K$, dual to the inclusion $K[x_1,\ldots,x_n]\to K^a[x_1,\ldots,x_n]$. The closed points of $\mathbb A^n_K$ are the images of the closed points of $\mathbb A^n_{K^a}$.
Here are a few facts surrounding/interpreting that geometric description :
a) A prime ideal $\mathfrak p\subset K[x_1,\ldots,x_n]$ is maximal $\iff $ the extension $K\subset Frac(A/\mathfrak p)$ is finite.
b) An ideal $I\subset K[x_1,\ldots,x_n]$ is maximal $\iff $ there exists $a\in (K^a)^n$ such that $I$ is the zero set $I_a= \lbrace f\in K[x_1,\ldots,x_n]\mid f(a)=0\rbrace $ c) Given $a,b \in (K^a)^n$ , the corresponding maximal ideals $I_a, I_b$ are equal $\iff$ there exists a $K$-automorphism $s\in Aut(K^a/K)$ such that $s(a_i)=b_i$. (Note that $K^a/K$ is not assumed Galois.)
Geometrically b) says that $\mathbb A^n_{K^a} \to \mathbb A^n_K$ is surjective and c) describes the fibers of that morphism as the orbits of $Aut(K^a/K)$ acting on $(K^a)^n$