I would like to understand this concept. It seems to be important (for the theory of perverse sheaves), yet I don't know any nice exposition of the properties of smooth sheaves.



A nice explanation of what is a smooth (l-adic) sheaf?
I'll give an answer, only because I'm interested in chasing down these references
myself. But all I'm doing is assembling references. I assume that
BCnrd will keep me honest. [July 21: I've added some remarks about constructibility, to
makes this more useful (at least to me).]


Since I'm a complex geometer rather an arithmetic one, let me start with the first case for intuition. If $X\_{an}$ is a (connected) complex variety endowed with the classical topology then one knows that representations of the usual $\pi\_1(X\_{an},x)$ correspond to locally constant sheaves on $X\_{an}$. This is classical. A good source of examples are as follows:
if $f:Y\to X$ is a surjective smooth proper map, then it is topologically
a fibre bundle (Ereshmann). Therefore $R^if\_\*\mathbb{Z}$ is locally constant. The 
corresponding $\pi\_1(X)$-module is the monodromy representation. The most
general statement one can make, without making any assumptions
on $f$, is that the proper direct image $R^if\_!\mathbb{Z}$ is constructible.
Note that constructibility can mean different things in the topological world.
The best notion (from my point of view) is what is sometimes called algebraic constructibility: there exists a partition of the base into Zariski locally closed strata
such that the restrictions of the sheaf are locally constant. The only reference that I know which takes this viewpoint is Verdier, *Classe d'homologie associée à un cycle*. If people are aware of other sources, please let me know.


Remarkably, the analogous results hold in the $\ell$-adic case, although for different reasons. Let $X$ be variety over some field.
A lisse (resp. constructible) $\ell$-adic sheaf is now a prosheaf 
$$\ldots \mathcal{F}\_n\to \mathcal{F}\_{n-1}\ldots $$
on the etale site $X\_{et}$ such that each item above is a locally constant (resp. constructible) $\mathbb{Z}/\ell^n$-module etc. (see Freitag-Kiehl, pp 118-131, for the precise conditions). For lisse sheaves, each $\mathcal{F}\_n$ gives a representation of the etale fundamental group 
$$\pi\_1^{et}(X,x)\to GL\_N(\mathbb{Z}/\ell^n)$$
($x$ a geom. pt.). So passing to the limit, we get a continuous representation
$$\pi\_1^{et}(X,x)\to GL\_N(\mathbb{Z}\_\ell)$$
This constuction is an equivalence [FK,p 286].


The corresponding result that $R^if\_\*\mathbb{Z}\_\ell$
is lisse, when $f$ is smooth, proper and surjective, should follow from Theorem 20.2 of Milne "Lectures on etale cohomology" from his website. The contrucibility of $R^if\_!\mathbb{Z}\_\ell$ would follow from SGA4 exp XIV 1.1 (It ought to be in [FK,M], but I probably didn't look hard enough.)


When $X$ is defined over $\mathbb{C}$, one can compare cohomology for the classical
and etale topologies with general coefficients by applying SGA4 exp XVI 4.1 and taking inverse limits. A more
general comparison result for the "6 operations" is given in [Beilinson-Bernstein-Deligne
p 150], but the proof seems a bit sketchy.
Remark added July 22: Unfortunately, this part of the story appears to be inadequately 
addressed in the literature. See BCnrd's comment below.


