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.