The Nisnevich topology on $Sch$ is a Grothendieck topology strictly finer than the Zariski topology, and the etale topology is strictly finer than the Nisnevich topology.

Colin McLarty asked me for an example of a Nisnevich cover which is not a Zariski cover. The standard example I have seen in several places is rather of the distinction between etale and Zariski. Namely, a family of etale covers $\{A^1 - \{0\} \stackrel{(-)^2}{\to} A^1, A^1 - \{a\}\hookrightarrow A^1\}$ of the affine line $A^1$ over a field $k$ indexed by elements $a\in k^\times$, such that a cover is only Nisnevich if $a$ has a square root in $k$. I'm looking for something that will separate Zariski from Nisnevich.

Examples inspired by arithmetic or geometry are both good.

neverNisnevich since the extension of residue fields at the generic point is $k(x^2)\hookrightarrow k(x)$, which isn't an isomorphism. – Anton Geraschenko Jul 27 '12 at 6:07