related by the Dold-Kan correspondence
In the context of homological algebra derived functors are traditionally considered on a model structure on chain complexes and often they are evaluated only on chain complexes that are concentrated in a single degree. If instead they are evaluated on general chain complexes, one sometimes speaks of hyper-derived functors for emphasis.
For more see at derived functor in homological algebra.
If abelian sheaf cohomology is considered in terms of the derived functor of the global section functor, then the corresponding hyper-derived functor is hypercohomology. This, too, is really just the basic definition of (abelian) cohomology, but not restricted to Eilenberg-MacLane objects concentrated in a single degree.