TITLE: The "need" for cohomology theories
QUESTION [21 upvotes]: In many surveys or introductions, one can see sentences such as "there was a need for this type of cohomology" or "X succeeded in inventing the cohomology of...".
My question is: why is there a need to develop cohomology theories ? What does it bring to the studies involved ?
(I have a little background in homological algebra. Apart from simplicial homology and the fact that it allows to "detect holes", assume that I know nothing about more complicated homology or cohomology theories)

REPLY [10 votes]: The point is that different cohomology theories are applicable in different situations and are computed from different data. For example, simplicial/singular cohomology is computed from a triangulation (or the map of a simplex) into your space, while, for example, Cech cohomology is computed from just the different open covers of your space.
Also, for certain algebraic-geometry applications, no matter how we define the topology on a variety, we never seem to get enough open sets to do any kind of homological computations with, and so the development of etale cohomology solved this obstacle and let us use easy to compute 'topological' tools to investigate geometric objects.