TITLE: What are normal schemes intuitively?
QUESTION [24 upvotes]: A ring is called integrally closed if it is an integral domain and is equal to its integral closure in its field of fractions. A scheme is called normal if every stalk is integrally closed.
Some theorems on normality:

A local ring of dimension 1 is normal if and only if it is regular.
(Serre's criterion) A scheme is normal if and only if it is nonsingular in codimension 0 and codimension 1 and every stalk at a generic point of an irreducible closed subset with dimension $\ge 2$ has depth at least 2.
Every rational function on a normal scheme with no poles of codimension 1 is regular. 
(Zariski connectedness): If $f:X\rightarrow Y$ is a proper birational map of noetherian integral schemes and $Y$ is normal, then every fiber is connected.
Normal schemes over $C$ are topologically unibranched.

But the proofs I've seen are fairly ad-hoc, and I was wondering if there's some geometric perspective that would clarify these results. The only result here thats an "iff" is Serre's criterion, but I don't understand depth geometrically so I'm not sure how to interpret it.
Is there some nice geometric perspective on normality?

REPLY [15 votes]: In the world of locally Noetherian schemes, Serre's criterion can be made quite geometric.
Let $X$ be a reduced, locally Noetherian scheme.  Then...

$X$ is $R_1$ iff the singular locus has codimension at least 2.
$X$ is $S_2$ iff, for each $Y\subset X$ of codimension at least $2$, the regular functions on the complement $X-Y$ extend to regular functions on $X$.

This second fact can be found in Ravi Vakil's notes (Theorem 12.3.10), or in this MathOverflow post.
Roughly speaking, normalizing a variety improves singularities as follows.

In codimension $1$, it completely resolves them.
In codimension $\geq 2$, it improves them enough so that rational functions defined on their complement can be extended to the singularity.