Lurie has a discussion about hypercoverings vs coverings, and sheaves, in Higher topos theory.
Artin-Mazur LNM0100 defines hypercovering on page 96. Def: A hypercovering of a pointed site C is a pointed simplicial objects with values in the site, such that two conditions are satisfied. A hypercovering of an object X is a hypercovering of the site C/X. Example: Take C to be the site of sets, then a hypercovering is a simplicial set which is nonempty, Kan, and contractible. Some intuition explained, and some lemmas. Cor 8.15 spectral sequence ass to a hypercovering and an abelian sheaf.
Pal said a hypercovering is a simplicial object in the cat of coverings, which is Kan contractible, i.e. trivially fibrant.
nLab page on Hypercovering