A standard fact about intersection multiplicities of algebraic plane curves is that, for curves $X=V(F(x,y))$ and $Y=V(G(x,y))$, if $p\in X\cap Y$ is a non-singular point of $X$ and $Y$ then
$$ I_p(X,Y)=1 $$
if and only if the tangent lines of $X$ and $Y$ at $p$ are distinct. In other words the curves have $0$-th order contact at $p$ (if $F_y(p)\neq 0$), where two curves have $k$-th order contact at $p$ if they share the point $p$ and the $k$ derivatives of $y$ with respect to $x$ at $p$ agree.
I believe it's true that
$$ I_p(X,Y)=k $$
if and only if the curves $X$ and $Y$ have $(k-1)$-th order contact at $p$, assuming $p$ non-singular and that the curves do not have $k$-th order contact. For small values of $k$, one can show this by locally parameterizing $X$ about $p$ using a Taylor series, and then considering the valuation of $G$ evaluated at this local parameterization.
Does anyone know where I can find a reference for this fact, or if this is not true?