Let $C$ be a projective irreducible non-singular curve over a field $k$ and let K be its function field. It applies that $(k[X,Y]/I(C))_{(X-a,Y-b)}$ (i.e. the localization of $k[X,Y]/I(C)$ at $(X-a,Y-b)$) is a discrete valuation ring w.r.t. $K$ and $k$, where $(a,b)$ is a point on $C$. My question is now:
Can any discrete valuation ring w.r.t. $K$ and $k$ linked to $C$?
(maybe to some points or a prime ideal of $k[X,Y]/I(C)$)?
Note that if $k$ is algebraic closed any discrete valuation ring is a local ring of some point of $C$.
Suppose that $C/k$ is an integral projective curve. Then, there is a bijection
$$\left\{\text{points of }C\right\}\longleftrightarrow\left\{\begin{matrix}\text{discrete valuations}\\\text{of }K(c)\\ \text{which are trivial}\\ \text{on }k\end{matrix}\right\}$$
The mapping being, as you indicating, sending $p$ to the valuation $v_p$ which is "order of vanishing of a function at $p$".
This is an application of the valuative criteria for properness.