TITLE: Is there a "geometric" intuition underlying the notion of normal varieties?  
QUESTION [69 upvotes]: I first got concious of the notion of normal varieties around 3 years ago and despite the fact that by now I can manipulate with it a bit, this notion still puzzles me a lot.
One thing that strikes me is that the definition of normality is so entirely algebraic.
From my common sense understanding the notion of normal varieties restricts the class of spaces that we consider to more-less reasonable ones. It looks to me that this definition is analogous to the definition of pseudo-manifold. At least the obvious similarity is that in both cases the set of non-singular points is connected.
Normality pops up everywhere and its definition is very short. But it is hard for me to imagine that a differential topologist or differential geometer could come up with such a definition.
Why is the notion of normatilty is so omnipresent? 
What is "geometric" meaning of normality?  
Maybe a more concrete question would be like this. Suppose $X$ is an irreducible algebraic subvariety in $\mathbb C^n$ with singularities in co-dimension $2$. Can one somehow looking on singularities, their stratification and the way $X$ lies in $\mathbb C^n$ say if it is normal or not? 
Added. Who was the person who invented this notion? 
I would like to thank everybody for useful comments and links.

REPLY [13 votes]: An excellent non-algebraic meaning (using analysis) of normality is found in Kollar's article in the Bulletin of AMS (1987).
Restrict to irreducible varieties $X$ so we can talk of function fields.
A point $x_0\in X$ is considered normal whenever a rational function exhibits decent behaviour in a neighbourhood of $x_0$ then it finds a place in the local ring of $X$ at $x_0$.
Decent behaviour here is: If $f\in K(X)$  and if  $|f(x)|$ remains a bounded function  as $x$  approaches $x_0$ by  paths lying in $X$, then $x_0$ should be good enough to admit $f$ in its local ring. 
This survey article of Kollar is about Mori's Fields-medal winning work on 3-folds. But it starts from the scratch defining what  an algebraic variety is. It is a great  source to learn the meanings of fundamentals objects of algebraic geometry. (for example  Kollar explains why we have to deal with line bundles when we study projective varieties).