Urs Schreiber has already addressed parts of this question [here][1]. There's lots of good stuff to mine through in the nLab entry on [cohomology][2].


  [1]: https://mathoverflow.net/questions/4214/equivalence-of-grothendieck-style-versus-cech-style-sheaf-cohomology/5815#5815
  [2]: https://ncatlab.org/nlab/show/cohomology