Geometry question: Regular finite map of algebraic subsets The following is a definition from Milne's a.g. Notes.
Let $\phi:V\to W$ be a regular map of algebraic subsets. The map is finite if $k[V]$ is a finite $k[W]$  algebra. 
What exactly should I be thinking geometrically when you have a finite regular map?
Thanks for all and any help.
 A: The definition should be -- the map is finite if $k[V]$ is finite as a $k[W]$ module. Not as an algebra -- the difference is crucial! (And confusing at first.)
If $k[V]$ is finite as a $k[W]$ algebra (finitely generated as an algebra), then we say the map is of finite type. All maps between varieties are finite type -- roughly, it corresponds to having finite dimensional fibers. 
It may be helpful to keep in mind that a morphism is finite iff it is proper and has finite fibers. So, when it comes to the point that you are dealing mainly with projective varieties (in which case all of the morphisms are "proper"), you can check to see if the fibers are finite to see if a morphism is finite. So, in particular all non-constant morphisms between projective (irreducible) curves will be finite.
It is useful to keep the following example in mind: If you consider the inclusion of $A^1 \setminus {0}$ into $A^1$, this morphism is not finite, even though the fibers are always points. (You should check these claims.)
See here:
https://mathoverflow.net/questions/41390/morphism-between-projective-varieties
https://mathoverflow.net/questions/1634/finite-type-finite-morphism
