Roughly, Noetherian schemes are (locally) given by finitely many equations, and this makes it possible to make inductions on the number of equations. Also, it fits very well to the classical intuition for algebraic varieties.
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.