TITLE: Examples of morphisms of schemes to keep in mind?
QUESTION [61 upvotes]: What are interesting and important examples of morphisms of schemes (especially varieties) to keep in mind when trying to understand a new concept or looking for a counterexamples?
Examples of what I'm looking for:

The projection from the hyperbola to the affine line has finite fibers but it is not a finite morphism
The Froebnius map is a morphism of varieties which is a bijection nevertheless not an isomorphism
The map $(x,y) \mapsto (xy, y)$ shows that the image of an affine variety need not be affine

Thanks for the excellent answers! To be a little more specific, I am especially interested in reasonable examples (so no line with infinitely many origins or the product of infinitely many fields). If the examples already make sense in the classical settings then all the better.

REPLY [3 votes]: Since my list of examples is becoming too long to be easily  legible, let me start a new one.  
Let $k$ be an algebraically closed field of characteristic $2$ and $C\subset \mathbb P^2_k$ be the conic $x^2-yz=0$ .
Identify the $y$-axis $V(x)$ with $\mathbb P^1_k$ through $(0:y:z)=(y:z)$, project $C$ onto $V(x)$ from the point $A=(1:0:0)$ and consider the projection  morphism $$p:C\to \mathbb P^1_k: (x:y:z)\mapsto (y:z)$$  That morphism is bijective but ramified at every point of $C$ i.e. its tangent map is zero everywhere!
  Geometrically all lines joining $A$ to a point $P\in C$ are tangent to $C$ at $P$.
 The conic $C$ certainly deserves its  technical denomination of "strange curve" !