Equivalent definitions for normal variety Can anyone show (or provide a reference) that an algebraic variety $X$ is normal iff every finite, birational morphism $f:Y\rightarrow X$ is an isomorphism. 
More importantly, can you describe your intuition that connects these two viewpoints.

There are a few things going on here.  First of all, we should say something like "isomorphism with an open subset".  Then one implication is just a form of what is usually called Zariski's main theorem.
The other implication follows from normalization itself.  If $X$ is not normal, then $\tilde{X}\to X$ is a finite, surjective, birational map which is not an isomorphism.
As for intuition, I would say that I always associate normality with regularity in codimension $1$ (this is part of Serre's criterion*).  Since a birational morphism is an isomorphism away from a set of codimension $1$, it seems reasonable that we would first look to normality when deciding if such a morphism can be extended.
* Specifically, a variety is normal if and only if its singular locus has codimension at least $2$, plus an additional technical condition (it is enough to be a hypersurface in something smooth).
