Cohomology classes are exact forms which are not closed. If you integrate a function along a closed surface, the integral depends only on the cohomology class of the integrand. Loosely, I tend to hear that cohomology measures obstructions to creating global solutions from local solutions, say of an equation dx=y. The cohomology class of y measures how this equation fails to have a global solution.

I think that cohomology is less geometric than homology, but the ring structure makes it more useful in many situations.