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 = $\text{Spec } k[x^2, xy, y]$.



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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][1]

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.


  [1]: https://mathoverflow.net/questions/186406/obtaining-non-normal-varieties-by-pushout/186650#186650