What does henselization do in etale cohomology theory? When I asked my professor they say:

A polynomial of a ring gives an etale cover of the spectrum of that ring, and the descent with respect to this cover is the henselian codition.

How to interpret this in detail? Can I find a reference for this in Fu Lei's Etale Cohomology Theory?
 A: In the Zariski topology, the local ring at a point is always local, and we have an adjunction between schemes and locally ringed spaces. In the étale topology, the local ring at a point is always strictly Henselian (Henselian with separably closed residue field), and we have an adjunction between Deligne-Mumford stacks and strictly Henselian ringed topoi. Assuming the topos has "enough points", this simply means that the local rings of all its points are strictly Henselian.
Now, let's look at this geometrically. The strictness condition is basically the same as considering geometric points instead of closed points, so let's focus on Henselian-ness. For a local ring $(R,\mathfrak{m})$, being Henselian  is equivalent to the condition that any finite extension of $R$ decomposes into a product of domains. What does this look like geometrically? Failure to be a domain looks like irreducible components (e.g. subvarieties) intersecting at $R$. Being Henselian means that you can "separate" these components into their separate pieces. Essentially, it's an algebraic version of what complex analysts would call a "ramified cover". This is just an étale cover, and that's the connection: the local ring at a point of a scheme (over a separably closed field) is Henselian iff the branches at that point (which can be separated using étale covers) locally split as a product.
What your professor may have meant by "a polynomial gives an étale cover" is that $R[x]/(f)$ is an étale extension of $R$, and the "descent condition" is that the "splitting" in the cover yields an actual decomposition into a product, locally.
