If I understand correctly:
Given an algebraic curve $C$ and a point $P \in C$, the stalk at $P$ is always a local ring.
The stalk at $P$ is a discrete valuation ring iff $P$ is a non-singular point of $C$.
Now "valuation ring" is an intermediate notion between "local ring" and "discrete valuation ring." But I'm not sure what it means geometrically.
Question. What do valuation rings mean geometrically?