Suppose $k$ be a field and we consider polynomial ring over $k$, $k[x_1,x_2]$. Let $f\in k[x_1,x_2]$. We define a (non Archimedian valuation) map $v:k[x_1,x_2]\longrightarrow \mathbb Z\times \mathbb Z$ (here $\mathbb Z\times\mathbb Z$ has lexicographic order)in the following way: Suppose $f=a_0+a_1x_2+\cdots +a_nx_2^n$, where $a_i\in k[x_1]$ and $a_n\neq 0$. We define
$v(f)=(v_{x_1}(a_n),n)$, where $v_{x_1}$ is defined by the degree of $a_n$.
We can also define another valuation on $k[x_1,x_2]$ by writing $f$ as polynomial in $x$ and defining similar way.
My question is what are the possible valuations on $k[x_1,x_2]$? Is there a general procedure to determine all possible valuations on $k[x_1,x_2,\cdots ,x_n]$?
No, I suspect that there must be far too many valuations to have any hope of a simple classification like that. For example there are the orders of vanishing along all kinds of varieties in the plane, including infinitely near things (i.e., orders of vanishing of derivatives; or, vanishing along blowups). And that is just barely getting started. Now, I don't really know a whole lot about valuation theory, but that's my guess, that there are far too many valuations and kinds of valuations to have any hope of classifying them.
Perhaps you might be interested in this paper that classifies all the valuations on the power series ring $\mathbb{C}[[x,y]]$ and shows that the set of valuations has a natural tree structure (!!).