$H^1$ has an interpretation as torsors. But what about the higher $H^i$ (in the setting of algebraic geometry, and étale or flat cohomology)? For example $H^2(X, \mathbf{G}_m)$ is (often) isomorphic to the Brauer group, i.e. equivalence classes of Azumaya algebras.
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Depending a bit on your context, try $n$gerbes. That should handle all the usual cases. Look at papers by Larry Breen and more recently Aldrovandi and Noohi in Advances (Butterflies is the keyword to look for!). 


I would say have a look at Duskin's paper '$K(\pi,n)$torsors and the interpretation of "triple" cohomology' (pdf) and his student Glenn's paper 'Realization of cohomology classes in arbitrary exact categories' J. Pure Appl. Algebra, vol. 25, (1982) pp. 33105. Also Duskin's 'Higher dimensional torsors and the cohomology of topoi : The abelian theory' Lecture Notes in Mathematics, Volume 753 (1979), pp255279. I'm afraid you have to move beyond just working with schemes to simplicial schemes and whatnot. 

