Pulling back divisors to the generic fiber of an elliptic surface

Question

Let $S \rightarrow C$ be an elliptic surface over an algebraically closed field $k$. Let $\eta$ be the generic point of $C$ and $E = S_\eta$ the generic fiber. We thus have a natural map of schemes $E \rightarrow S$ from the universal property of the fiber product. How can I think about pulling divisors of $S$ back to $E$ via this map? For instance, what happens to a vertical divisor, a section, a more general horizontal divisor, etc. I'm not very well-versed in schemes, so basic answers are greatly appreciated, including answers that put $S$ and $E$ in Weierstrass form for concreteness.