Look at the following lines I found in the book "Moriwaki - Arakelov geometry" (beginning of chap. 4):
Let $S$ be a connected Dedekind scheme with function field $K$. Let $\pi:X\to S$ be a projective flat morphism of relative dimension 1. Assume that $X$ is regular and that $\pi$ is smooth over $K$.
I know the definition of a smooth morphism between two schemes, but what is the smoothness over $K$ (the function field of $S$)?
edit: I know the following definition of smoothness from Liu's book:
$f:X\to S$ of finite type is smooth at $x\in X$ if it is flat at $x$ and moreover if $X_{f(x)}\to\text {Spec } k(f(x))$ is smooth at $x$. Then, a scheme $Y$ over a field $k$ is smooth at $y\in Y$ if the points in the base change $Y_{\overline k}$ lying over $y$ are regular.
It means that the generic fiber is smooth.
Explicitly, you have the generic point $Spec(K)\to S$, so you can take the pull-back $\pi_K: X_K = X\times_S Spec(K)\to Spec(K)$, and you ask that this is a smooth morphism.