Maybe it is well known to experts or maybe it is just a stupid idea, but I will ask any way.
We know that if $X$ is a topological space, then there is an equivalence of categories between the category of locally constant sheaves (of sets) on $X$ and the category of covers (sous-entendu local homoemorphism) of $X$.
The equivalence is given by "$\Rightarrow$" using the "espace \'etal\'e" of sheaves, "$\Leftarrow$" taking the sheaf of sections.
Now I replace $X$ by a scheme (locally noetherien or something?), and I think there is an equivalence of categories between the category of locally constant constructible sheaves of sets (By constructible I mean the constant values should be finite) on the \'etale site of $X$ and the category of finite \'etale coverings of $X$.
I tried to construct the functor "$\Leftarrow$": Given $Y\rightarrow X$ finite \'etale, we associate to any $T\to X$, the set of sections $T\to Y\times_XT$. This is a locally constant constructible sheaf if $X$ is locally noetherian: You decompose $X$ into connect components and by SGA1 corollary 5.3 then you can see easily that on each connected component the association is a constant sheaf with a finite constant value.
Is this an equivalence? If it is how one constructs the quasi-inverse?
Furthermore, do we have any formulation like the finite representations of $\pi_1^{\text{et}}(X,x)$ is equivalent to the category of locally constant constructible \'etale sheaves (of vector spaces) on $X$. If this is true it should be a direct consequence of Grothendieck's main theorem on $\pi_1^{\text{et}}$ and the above statement.
Best Answer
Consider your functor from étale coverings to locally constant constructible sheaves. It is fully faithful, by Yoneda's lemma. The fact that it is essentially surjective follows from descent theory. If $F$ is a locally constant constructible sheaf, take an étale cover $\{U_i \to X\}$ such that the restriction of $F$ to $U_i$ is constant; call $A_i$ a finite set such that $F_i := F\mid_{U_i}$ is represented by $U_i \times A_i := \bigsqcup_{a \in A_i}U_i$. The sheaf $F$ gives descent data $\mathrm{pr}_2^*F_j \simeq \mathrm{pr}_1^*F_i$ on the fibered products $U_i \times_X U_j$; by faithful flatness, these give descent data for the covers $U_i \times A_i \to U_i$, yielding a finite étale cover of $X$ that represents $F$.
[Edit] I should have pointed out that descent for étale covers works because étale covers are affine maps; ultimately, this relies on descent for quasi-coherent sheaves, a version of which is used in Scott's answer.