If you fix a grothendieck topology $\tau$ on a category $C$, and you take $U$ a simplicial presheaf on $C$, and let $V$ be a discrete simplicial presheaf coming from a representable sheaf on $C$, then $U\to V$ is a hypercover if at each level $U_n$ is a coproduct of representables, and additionally $U_0\to V$ is a $\tau$-cover, and each $U^{\Delta^n}\to U^{\partial\Delta^n}$ is a $\tau$-cover when restricted to degree zero (simplicial exponential objects)

Then taking a bousfield localisation of sPre(C) with respect to these maps gives us a category of spaces which are said to satisfy $\tau…

Well that condition above gives us 'homotopy sheaves of spaces', so it says something like 'we can lift sections correctly up to homotopy'

In the particular case of the nisnevich site, one can check Nisnevich-fibrancy on a special family of pullback squares.

A more general in one sense, less general in another sense answer: When computing cech-cohomology for the zariski site, cech and sheaf agree in all degrees on a sufficiently nice scheme (say noetherian separated) *for a quasi-coherent sheaf*. For the etale site cech and etale agree for *any* etale sheaf while requiring much less from …