Cohomological dimension arises in different contexts, algebraic topology, algebraic geometry, group theory... So I think that it would be difficult to give a single reference for this broad topic. 

It was already mentioned that Brown "cohomology of groups" has a chapter on it, and I think this is a good reference for cohomological dimension of groups. However cohomological dimension  also arises in the context of topology, 
and when the space is well-behaved, say a manifold, it just coincides with the usual topological dimension. So it is perhaps better to start building intuition in that context.


Topological dimension is defined with covers, so Cech cohomology, which is also defined using covers, is perhaps the best cohomology to start with in order to understand the relation between the two notions. Cech cohomology is used for practical computations in algebraic geometry, so it will be useful if you are interested in that subject.
There is a chapter on Cech cohomology in Hartshorne, but there may be better references more focused on dimension in the algebraic setting. If you are interested in the differential viewpoint, then Bott-Tu "Differential forms in algebraic topology" is a good reference.

Bredon, "sheaf theory", has  a chapter on dimension from the topological viewpoint. Here is what I learnt from it. You may expect that if X is a well-behaved subset of Y, the cohomological dimension of X (that is, the greatest integer, or $\infty$) is less than or equal to the dimension of Y. This is true for some cohomologies, like the sheaf cohomology with constant coefficients (X,Y compact), but it is false for others. In particular, singular cohomology behaves a bit erratically with respect to cohomological dimension. Barratt and Milnor built in 62 an example of a compact manifold in R^3 (infinitely spheres of radius 1/n tangent to xy plane at the origin) with infinite singular cohomological dimension. This is definitely surprising !