To be more precise, a more fundamental concept is that of a morphism of finite type $X \to S$. For a fixed $S$, these make up a category, but it is not so well behaved and there are quite some subcategories such as the category of abelian schemes which behave only well if you assume that $S$ is, say, locally noetherian. On the other hand, you can also consider the category of morphisms of finite presentation $X \to S$ and it turns out that quite often this is more natural since now you can drop any finiteness conditions on $S$. The reason is, roughly, that here $X$ is described by finitely many equations, and that these equations also satisfy only a finite number of relations, the latter being important for reduction and induction arguments in the spirit of the Five Lemma. If $S$ is noetherian, we get the latter for free and that is the reason why you often just assume it.