ag.algebraic geometry - what notions are "geometric" (characterized by geometric fibers)?

Sorry The title might not be suggestive enough.

The question is about things like the following: A reductive group scheme is defined to be a (really nice) group scheme whose geometric fibers are reductive groups. So in some sense, "reductiveness" is some kind of "geometric" notion.

So whatelse properties of schemes can be checked only on geometric fibers? What I know, for example, given a scheme over a field $k$, it is projective iff it is projective over $bar{k}$. But can this be extended to any base scheme?

In particular, is there any reference that collects such results? And "WHY" should this work? for example, WHY geometrically-reductive group schemes turn out to be the right generalization of reductive algebraic groups?

Sorry this question might be a little to vague, and thank you in advance.