Geometric view of rings Intuitively, a map is determined on a region by its values on the points of a region.  In the language of affine schemes, this suggests that $f(\mathfrak p_x)$ is determined by $\{f(\mathfrak m)\colon \mathfrak p_x\subset \mathfrak m\}$ (where we write $f(\mathfrak q):= f\bmod\mathfrak q$).  But this isn't true, as can be seen by considering a local ring.  Why is my intuition wrong?
($A$ is a commutative ring, $f\in A$, $x=\mathfrak p_x\in\operatorname{Spec}(A)$, $\mathfrak m$ is a maximal ideal of $A$)
 A: You need to look more generally than maximal ideals or even prime ideals if you want a notion of "point" of an affine scheme that faithfully describes morphisms. Namely, for an affine scheme $\text{Spec } S$, an $R$-point is a morphism $\text{Spec } R \to \text{Spec } S$, or in the opposite category a morphism $S \to R$. Thinking about maximal and prime ideals only gives you information about $k$-points for $k$ a field, and this isn't enough in general (for example if $R$ has nonzero nilradical).
$R$-points should be thought of as "generalized" or perhaps "fat" points. For example, a $k[t]$-point is a one-parameter family of $k$-points, and a $k[t]/t^2$-point is a $k$-point together with a tangent vector. 
It turns out to be the case that a morphism of affine schemes is completely determined by what it does to $R$-points for all commutative rings $R$. (This is actually trivial; just take $R = S$ and $S \to S$ the identity morphism.) 
A: Schemes carry much more information than their set of points, which is why a morphism of schemes consists of a map of structure sheaves together with a continuous map of the underlying topological spaces. Consider, for example, $\text{Spec } \mathbb{C}$. The spectrum of any field has the same underlying point set (namely, a singleton), but we wish to distinguish between different fields. As for morphisms, there is only one map of spaces $\{ * \} \to \{ * \}$, but maps of schemes $\text{Spec } \mathbb{C} \to \text{Spec } \mathbb{C}$ are in correspondence with endomorphisms of $\mathbb{C}$ as an abstract field, and of course there are many of these.
