Let $S$ be a projective K3 surface, and let $C \subset S$ be an integral curve. Assume also, that all other curves in the linear system $|C|$ are integral, and let $n$ be the dimension of $|C|$.
According to Beauville's Counting rational curves on $K3$ surfaces, that linear system can be realized as a scheme $$ f: \mathcal{C} \to \mathbb{P}^n,$$ where the fibers $C_t = f^{-1}(t)$ are exactly the elements in $|C|$.
How does one construct $\mathcal{C}$?
Let $X$ be any variety and $|D|$ a linear system. Then one has the incidence variety $\Gamma\subset X\times |D|$, consisting of pairs, $(x,E)$ where $x\in E$. Then take the projection $\Gamma\to |D|$ to get what you want.