Essential geometric morphisms on the étale site.  Hi!
There is a risk that this question might be utterly trivial, and if it is, my sincerest apologies, but I haven't been able to find anything in the literature. 
Let $f:X \rightarrow Y$ be an étale morphism of schemes. This gives rise to a geometric morphism of topoi $f^\ast:Sh(Y) \rightleftarrows Sh(X) : f_\ast$, and my question is now:
In what cases do $f^\ast$ have a left adjoint? Are there any explicit descriptions for this for say affine schemes?
 A: The answer is always.  Let $\mathrm{Et}(X)$ and $\mathrm{Et}(Y)$ denote the étale sites.  There is a functor $f_! : \mathrm{Et}(X) \rightarrow \mathrm{Et}(Y)$ sending an étale $X$-scheme $p : U \rightarrow X$ to $f \circ p : U \rightarrow Y$.  This functor is cocontinuous (SGA4.III.2.1) and continuous (SGA4.III.1.1).  By SGA4.III.2.6, any functor that is both continuous and cocontinuous gives rise to a morphism of topoi $(f^\ast, f_\ast)$ where $f^\ast$ has a left adjoint.
A: As Jonathan said, the answer is always, and the left adjoint $f_!$ has a simple description as the "extension by $\emptyset$" functor. Think of an open immersion $j:U\rightarrow X$ of topological spaces, then we have the usual extension by $\emptyset$ functor by sheafififying the presheaf
$j_!\mathcal{F}(W)=\mathcal{F}(W)$ if $W\subset U$
$j_!\mathcal{F}(W)=\emptyset$ otherwise. 
It's an easy calculation to show that this is left adjoint to $j^*$.
Translating this to the étale topology, if $f:X\rightarrow Y$ is an étale morphism of schemes, we have the extension by $\emptyset$ functor by sheafififying the presheaf
$f_!\mathcal{F}(W\overset{f}{\rightarrow} Y)=\coprod_{g\in\mathrm{Hom}_Y(W,X)}\mathcal{F}(W\overset{g}{\rightarrow} X)$
and again it's fairly easy to verify that $f_!$ does what we want it to.
It's worth noting that to get an adjoint for abelian sheaves, we need to replace this coproduct by the coproduct in the category of abelian sheaves, i.e. direct sum. Thus $f_!\mathcal{F}$ depends on whether we are considering $\mathcal{F}$ as a sheaf of sets or of abelian groups.
It's possibly also worth noting that we don't get a geometric morphism $f^*:\mathrm{Sh}(Y)\leftrightarrows \mathrm{Sh}(X):f_!$ because $f_!$ doesn't preserve limits. Despite this, the version of $f_!$ for abelian sheaves is actually exact!
