M. Kashiwara and P. Schapira, Sheaves on Manifolds,
Grundlehren der Mathematischen Wissenschaften, Springer-Verlag,
The first two chapters of this book give an excellent
derived categories in general and of sheaves on a topological space,
but most of the rest of the book is beyond the scope of this course.
A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers,
in Analyse et topologie
sur les espaces singuliers I (Luminy, 1981),
This is the monograph in which the theory of perverse
was first developed. It's somewhat dense, but it's still the
best reference for all the relevant material.
R. Hartshorne, Algebraic
geometry, Graduate Texts in Mathematics, No. 52,
§II.1 of this book is a good brief
account of the basics of sheaves.