Let us say that $P$ is a property of a scheme (so it is going to be an object here) of being affine (that it is isomorphic to Spec $A$). Then the a morphism of schemes

$$\pi: R \to S$$

can be said to have this property if $\pi$ is affine. How do we know that? We can say that for every subscheme $U \in S$, the object $\pi^{-1} (U)$ should have the property $P$ (in this case, being affine). Using the Affine Communication Lemma, we can check it for every subset and verify the properties locally on the base scheme $S$.

Now, while it is similar to the saying that every fiber on $S$ should have property $P$, it is more than this fiber-by-fiber analysis. It behaves well when fiber moves. Such a fiber is more of a scheme-theoretic fiber. Such a change of perspective should be helpful, as I understand, in the moduli spaces (and deformation theory) discussion where there is a morphism of universal family, and its properties should determine the properties of the moduli space. A property will ensure that these fibers behave well when you move on the base moduli space.