TITLE: Intuition Behind, or Canonical Examples of Finite Type Morphisms
QUESTION [12 upvotes]: I'm new to the world of schemes, so I'm still trying to grasp some of the basics.  I understand most of the simple topological properties of schemes, as well as some of the sheaf-theoretic properties like reducedness, integrality, normality, etc.  But I'm having a hard time digesting what finte and finite type morphisms really entail concretely.  Also, Hartshorne doesn't seem to include many nice examples from what I can tell.
Pretty much the only example I have down at this point is what it means for a scheme $X$ to be of finite type over a field $k$.  In such a case, we'd have a morphism of schemes $f:X \to \rm{Spec}(k)$, and since the spectrum of a field is simply a point, the inverse image is all of $X$.  Hence, just applying the definitions, we see that $X$ has a finite open cover by affine schemes $\rm{Spec}A_{i}$, where each $A_{i}$ is a finitely generated $k$-algebra.  If we add integrality of $X$, we observe that a variety over $k$ is simply a integral scheme of finite type over a field $k$.
So this is nice, but beyond this I'm failing to grasp any intuition.  Does anyone have any canonical examples that I should work through?  Or maybe any pointers on how to think about finite and finite type.  Eisenbud and Harris point out that pretty much all geometrically interesting morphisms are usually in these two regimes.  For example, I guess spectra of local rings are "non-geometric" as they aren't of finite type over a field.  This is a great example of something I don't understand well enough yet.

REPLY [2 votes]: I suggest reflecting upon the following themes.
Consider, say, an affine scheme of finite type $X = \text{Spec}\,A$ over an algebraically closed field $k$. In the sense of classical algebraic geometry, its points are in bijection with maximal prime ideals.
But in the schematic environment, its points are all prime ideals, for example $(0)$, if $A$ has no zero divisors. This $(0)$ is called the generic point of $X$.

Describe all points of the closure of the set, consisting of one point.
Show that this closure is an affine scheme itself. What is its function ring?
In particular, localization $\text{Spec}\,A_\mathfrak{p}$ generally has more points than just one closed and one generic point! There are generic points of other localizations: which ones?

This was about zooming.

But you can also start, with say, one closed point, and then successively extend its "neighborhood", but keeping it infinitesimally small: consider spectra of $A/(\mathfrak{p}^n)$, $n = 1, 2, 3, \ldots$.