A derived functor is a functor constructed from an additive functor using resolutions. Higher derived functors of a left exact or right exact functor measure its failure to be exact.
Let 
be a covariant functor that is an additive
functor between Abelian categories. If 
has enough
injectives, its right derived functors are computed by applying 
to an injective resolution
and taking cohomology. Dually, if 
has enough projectives, its left derived functors are computed
by applying 
to a projective resolution and taking homology
(Stacks Project n.d.; Weibel 1994). Ext and Tor are standard
examples.
