I'll give an answer, only because I'm interested in chasing down these references myself, but there's not much content. Anyone could do this. I assume that BCnrd will keep me honest.
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 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.
Remarkably, the analogous result hold in the $\ell$-adic case, although for different reasons. Let $X$ be variety over some field. A lisse $\ell$-adic sheaf is now a prosheaf $$\ldots \mathcal{F}_n\to \mathcal{F}_{n-1}\ldots $$ on the etale sie $X_{et}$ such that each item above is a locally constant $\mathbb{Z}/\ell^n$-module etc. (see Freitag-Kiehl, pp 118-131, for the precise conditions). 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 and proper, should follow from Theorem 20.2 of Milne "Lectures on etale cohomology" from his website.
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.