Total derived functor is an exact functor between triangulated categories (i.e. respects distinguished triangles and translation functor). Exactness of each individual $R^i F$ perhaps is not the best way to think about the situation.
From technical point of view when you try to calculate $R^iF$ you need a resolution, but you can choose any resolution because they are all quasi-isomorphic, this leads to a notion of derived category i.e. complexes of objects of a given abelian category up to quasi-isomorphisms. The definition has some technical difficulties and not very straightforward but you can find it in some textbooks on homological algebra. I like Weibel"s An Introduction to Homological Algebra. Derived category is an additive category but in general not abelian, it has structure of a triangulated category where notion of exactness is replaced by distinguished triangles and exact functors are functors between triangulated categories that respects distinguished triangles (and also translations).
Exact functor between abelian categories immediately gives you an exact functor between derived categories, but if you start with a non exact functor you can derive it and get an exact (total) derided functor between derived categories. Now remember that objects of derived category were complexes, in particular value of the total derived functor is a complex and you can take cohomology of that comlex and that is $R^iF$.
To sum up, there is a total derived functor between derived categories and it is exact, and (partial) derived functors $R^iF$ are cohomology if that functor.
Because we are in the algebraic geometry section I ought to mention one more reference where you can find discussion of this machinery with application to algebraic geometry: Hartshorne R. Residues and duality.