what is the simplest example of an etale cover which is not Galois? By "Galois morphism" I mean a morphism $f: Y \to X$ such that $Y \times_X Y$ is a disjoint union of schemes isomorphic to $Y$.
Let $X$ be reduced curve over afield of char. 0.
I wonder what is a simple example of an etale morphism $f: Y \to X$ which is not Galois in the above sense. 
update: What about a "geometric" example?  (i.e. the base field is algebraically closed).
 A: Let $C$ be a curve of genus $2$, say over $\mathbb C$.  Think of $C$ as a Riemann surface for a moment.  Then $\pi_1(C)$ is generated by four elements
$a,b,c,d$ satisfying the relation $[a,b][c,d] = 1$.   Let $p: \pi_1(C) \to S_3$ be the surjection that takes $a$ and $d$ to $(12)$, and $b$ and $c$ to $(123)$, and let $H \subset \pi_1(C)$ be the preimage under $p$
of a subgroup of order $2$ in $S_3$, so $H$ has index $3$ in $\pi_1(C)$ and is not normal.
Covering space theory shows that $H$ corresponds to a degree $3$ cover $C' \to C$
of Riemann surfaces which is not Galois.  Now the Riemann existence theorem
shows that $C'$ has a unique structure of algebraic curve over $\mathbb C$ so that
$C' \to C$ is an etale morphism, which will not be Galois.
This is the simplest sample in some strict sense: $\pi_1$ of a genus zero Riemann surface is trivial, while $\pi_1$ of a genus one curve is abelian (so all subgroups are normal), and all index two subgroups of a group are normal; thus we have to go to genus $2$ and a degree $3$ cover in order to find an example, and this is what I have done.  [Added: I should add that this is the simplest example if one wants an etale morphism of projective curves; Georges found a simpler example in his answer by considering non-projective curves.]
Of course, one could write down examples with explicit algebraic equations,
but I would have to begin with a genus $2$ curve, which is of the form
$y^2 = f(x)$ for some degree $5$ or $6$ equation, and then write down $C'$ explicitly.  By Riemann--Hurwitz, $C'$ has genus $4$, so I would then have to write down an equation for a genus $4$ curve, and find an explict degree $3$ map to $C$.  I haven't tried to do this; it's probably a good exercise, though.
A: You may take (over the characteristic zero field $\mathbb Q$)  the morphism $$f:Y=\mathbb A_{\mathbb Q(\sqrt [3]2)}^1 \to X=\mathbb A_{\mathbb Q}^1$$       corresponding to the inclusion $\mathbb Q[T] \hookrightarrow  \mathbb Q(\sqrt [3]2)[T]$   
Edit 
At Dima's request, here is a completely explicit geometric example. 
Let $A=\frac {\mathbb C[T,Z]}{(Z^3-3Z+2T)} $  and $g:Spec (A)\to \mathbb A^1_\mathbb C$ the étale morphism corresponding to the inclusion $\mathbb C[T] \hookrightarrow  A$    
Then $g$ is  not Galois because the field extension $\mathbb C(T)\hookrightarrow  K=Frac (A)=\mathbb C(T)(z) \;$ is not Galois since  the discriminant of $ \; Z^3-3Z+2T\;$ is $\;108(1-T^2)\;$ and it is not a square in $\mathbb C(T)$ .
However $g$ is not étale either since it is ramified over $T=\pm1$.
 The remedy is simple: delete these two points i.e. corestrict $g$ to $$X=A^1_\mathbb C\setminus \lbrace 1,-1\rbrace=Spec(\mathbb C[T,\frac {1}{T^2-1}])$$  and restrict $g$ to 
$$Y=g^{-1}(X)=Spec(B)$$
where $B=A[\frac {1}{T^2-1}]=\mathbb C[T,\frac {1}{T^2-1}](z)$.
Finally the required étale non Galois morphism is the restriction
$$f=res(g):Y\to X                  $$  
A final remark
Actually there exist  no non-trivial étale covers of the affine line $\mathbb A^1_k$ over an algebraically closed  field $k$ of characteristic zero . (The affine line is then said, unsurprisingly,  to be simply connected) .     
However in characteristic $p\gt 0$, surprisingly, there exist many étale covers of the affine line .
The simplest is the Artin-Schreyer cover $\; Spec \frac {\mathbb k[T,Z]}{(Z^p-Z-T)}\to \mathbb A^1_k=Spec (k[T]) \;$but there are loads more: this is still a topic of current research.
