Grothendieck's "Relative" Point of View I have often read that Grothendieck's insight was to put emphasis on studying the morphisms between schemes as opposed to just the schemes by themselves. What do we gain from this point of view? Why is it important that we study S-schemes, change of base, fibered products, and the like? Are there any specific concrete examples of this point of view in action?
 A: Here are some examples of the strength of this point of view, taken from Ravi chapter 18.
A classic theorem: The space of global sections of a coherent sheaf on a projective $k$ variety (regular functions, regular differentials, for example) is finite dimensional over $k$. (The finiteness of the space of differentials is one way that genus can be defined in algebraic geometry.)
A similar result holds in the relative situation, in the following way: If $X$ is a projective $A$-scheme, where $A$ is a Noetherian ring, and $F$ is a coherent sheaf on $X$, then $\Gamma(X, F)$ is finitely generated as an $A$ module.
A useful consequence of this is: A projective and affine morphism to a locally Noetherian scheme is finite (meaning, the induced maps on coordinate rings of affine patches are integral). More generally, the pushforward along a projective morphism to a locally Noetherian scheme sends coherent sheaves to coherent sheaves.
If you like, this is the relative version of the statement that affine and projective varieties over $\operatorname{Spec} k$ are finite over $\operatorname{Spec} k$, i.e. just a bunch of points.
A nice classical geometry fact follows from this: A morphism between two projective varieties with finite fibers is finite. 
The proof of this works in the relative projective space of $\operatorname{Spec}(A)$, and uses projective geometry intuition there (complement of a hyperplane being affine) to reach this conclusion by appealing to the aforementioned affine and projective maps are finite, which was also a consequence of the relative point of view.
