In the history of the subject they were introduced before the 1957 "Tohoku paper" of Alexander Grothendieck, which showed that the abelian category notion of injective object sufficed to found the theory.
The abstract framework for defining cohomology and derived functors does not need them.
Acyclic sheaves therefore serve for computational purposes, for example the Leray spectral sequence.
is a sheaf that is an injective object of the category of abelian sheaves; in other words, homomorphisms from
This result of Grothendieck follows from the existence of a generator of the category (it can be written down explicitly, and is related to the subobject classifier).
This is the case, for example, when looking at the category of sheaves on projective space in the Zariski topology.
The cohomology groups of any sheaf can be calculated from any acyclic resolution of it (this goes by the name of De Rham-Weil theorem).
A fine sheaf over X is one with "partitions of unity"; more precisely for any open cover of the space X we can find a family of homomorphisms from the sheaf to itself with sum 1 such that each homomorphism is 0 outside some element of the open cover.
Typical examples are the sheaf of germs of continuous real-valued functions over such a space, or smooth functions over a smooth (paracompact Hausdorff) manifold, or modules over these sheaves of rings.
can thus be computed as the cohomology of the complex of globally defined differential forms: A soft sheaf
This means that they are some of the simplest sheaves to handle in terms of homological algebra.