I first got concious of the notion of normal variety around 3 years ago and despite the fact that by now I can more-less manipulate with it, this notion still puzzles me a lot. What really strikes me is that this definition is so entirely algebraic.

From my pedestrian geometric point of view it looks that the notion of normal varieties should restrict 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.

This notion is obviously efficient (maybe even beautiful in its simplicity). But it is hard for me t imagine that a differential topologist or differential geometer could come up with such a definition. Why is the notion of normatilty is so efficient? 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 (or maybe on their stratification) say if $X$ is normal or not?