Is there a “geometric” intuition underlying the notion of normal varieties?

This is basically the same as roy smith's excellent comment, but I'd like to put a slightly different spin on it.

A normal variety is a variety that has no undue gluing of subvarieties or tangent spaces.

Let me explain what I mean by gluing. Given a variety $X$, a closed sub-scheme $Y \subseteq X$ and a finite (even surjective) map $Y \to Z$, you can glue $X$ and $Z$ along $Y$ (identifying points and tangent information). This is the pushout of the diagram $X \leftarrow Y \rightarrow Z$.

You might not always get a scheme (although you do in the affine case) but you always get an algebraic space. In the affine case, this just corresponds to the pullback in the category of rings.

Example 1: $X = \mathbb{A}^1$ glued to $Z = \bullet$ (one point) along $Y = \bullet, \bullet$ (two points) is a nodal curve.

Example 2: $X = \mathbb{A}^1$ glued to $Z = \bullet$ (one point) along $Y = \star = \text{Spec } k[x]/x^2$ a fuzzy point gives you a cuspidal curve.

Example 3: $X = \mathbb{A}^2$ glued to $Z = \mathbb{A}^1$ along one of the axes $Y = \mathbb{A}^1$ via the map $Y \to Z$ corresponding to $k[t^2] \subseteq k[t]$ gives you the pinch point / Whitney's umbrella = $\text{Spec } k[x^2, xy, y]$.

If I recall correctly, all non-normal varieties $W$ come about this way for some appropriate choice of normal $X$ (the normalization of $W$) and $Y$ and $Z$ (NOT UNIQUE). Roughly speaking, if you are given $W$ and want to construct $X, Y, Z$, do the following: Let $X$ be the normalization, let $Z$ be some sufficiently deep thickening of the non-normal locus of $X$ and let $Y$ be some appropriate pre-image scheme of $Z$ in $X$.

Edit: There is a proof available now HERE

Assuming this is true, you can see that all non-normal things are non-normal because they either have some points identified (as in 1 or 3) or some tangent space information killed / collapsed (as in example 2), or some combination of the two.

As J. C. Ottern has pointed out, Serre's criterion gives a clue. Normality is equivalent to $R_1 + S_2$. The interpretation of $R_1$ is easy: regular in codimension 1, the singular locus has codimension at least two, it is small, e.g. for a surface this means that the singular points are isolated. The interpretation of $S_2$ is the "extension property", every algebraic function defined on a open set whose complement is of codimension at least two, extends to all the variety. The proof of this fact is essentially contained in EGA IV$_2$ $\S$5.10.

At least in the case of complex algebraic varieties one can give a nice topological interpretation of the normality condition. Let us consider $V$ a complex algebraic variety, then its complex points $V(\mathbb{C})$ has the structure of a stratified pseudomanifold.

Let me recall that a stratified pseudomanifold $X$ is a filtered topological space $$X_0\subset\ldots \subset X_n$$ such that each stratum, i.e. a connected component of $X_i-X_{i-1}$ is a manifold of dimension $i$ and such that $X_{n-1}=X_{n-2}$ and such that the regular part $X_n-X_{n-2}$ is dense in $X$. Together with a local condition: the existence of conical charts.

Thus $V(\mathbb{C})$ comes equipped with such a geometric structure. In the setting of stratified pseudomanifold one has a notion of normal pseudomanifold and normalization is a fundamental concept in intersection homology. A pseudomanifold $X$ of dimension $n$ is said to be normal if for every point $x\in X$ the local homology group $H_n(X,X-x,\mathbb{Z})$ is isomorphic to $\mathbb{Z}$. Notice that a homological manifold is normal. Using Zariski’s Main Theorem, one can prove that a normal complex algebraic variety is a normal pseudomanifold.

If you consider a triangulation $T$ of $X$ ($dim(X)=n$) then you can also prove that $X$ is normal if and only if the link of eack simplex in the $n-2$-skeleton of $T$ is connected. This is proved in Goresky, MacPherson "Intersection Homology theory" (Topology Vol. 19 (1980)). In this paper the authors also explains how to build normalization topologically and how topological normalization satisfies a universal property. In the case of $V(\mathbb{C})$ its topological normalization in the sense of Goresky-MacPherson is homeomorphic to $V'(\mathbb{C})$ where $V'$ is the algebraic normalization of $V$.

Thus topologically normality corresponds to the connectivity of the links, the link of a point in an $n$-dimensional manifold being a $n-1$ sphere we see that topological normalization is the very first step to desingularization of stratified pseudomanifolds.

Here are two examples:

1) The pinched torus is not normal. It is a complex projective curve $C$ of equation $x^3+y^3=xyz$ in homogeneous coordinates $[x:y:z]$. It has a unique singular point $[0:0:1]$ and the link of this point $p$ is homeomorphic to two circles (we have $H_2(C,C-p;\mathbb{Z})\cong \mathbb{Z}\oplus \mathbb{Z}$).

2) The quadric cone is normal. It is an algebraic surface $S$ of equation $x^2+y^2+z^2=0$ in $\mathbb{P}^3(\mathbb{C})$ in homogeneous coordinates $[x:y:z:w]$ it has a unique singular point $[0:0:0:1]$. We notice that this space is homeomorphic to the Thom space of the tangent bundle of the $2$-sphere $S^2$. This remark gives a homeomorphism between the link of the singular point and the unit sphere bundle of the tangent bundle of $S^2$ which is connected (we get that $S$ is topologically normal).

Historicaly these two examples were important for our understanding of the failure of Poincaré duality for singular spaces, they appear in Zeeman's thesis:

E. C. Zeeman, "Dihomology III. A generalization of the Poincaré duality for manifolds", Proc. London Math. Soc. (3), 13 (1963), 155-183.

and also in McCrory's thesis:

C. McCrory, "Poincaré duality in spaces with singularities", Ph.D. thesis (Brandeis University, 1972)

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).

Regarding the question "Who was the person who invented this notion?", a paper of H. T. Muhly provides interesting background (as well as a geometric interpretation) for projectively normal:

In the terminology of the Italian School an algebraic variety is called "normal" if its system of hyperplane sections is complete. O. Zariski applies the term "normal" to an algebraic variety whose associated ring of homogeneous coordinates is integrally closed. The two concepts are not equivalent. Zariski refers to a variety which satisfies the former condition as "normal in the geometric sense" and to one which satisfies the latter condition as "normal in the arithmetic sense".

(...)

The object of this note is to characterize geometrically those algebraic varieties which are normal in the arithmetic sense. To this end we propose the following theorem: A necessary and sufficient condition that the $r$-dimensional algebraic variety $V_r$ be normal in its ambient projective space $P_n$ is that for every integer $m$ the linear system cut out on $V_r$ by the hypersurfaces of order $m$ in $P_n$ be complete.