The homotopy category of chain complexes over a ring has the same objects as the category of chain complexes and has chain maps modulo chain homotopy as its morphisms. Composition is induced by composition of chain maps. Its localization at quasi-isomorphisms gives the derived category.
More generally, a differential graded category 
has a homotopy category 
whose morphisms are the degree-zero cohomology
groups of its morphism complexes. For the differential
graded category of complexes, closed degree-zero morphisms are chain maps and
degree-zero boundaries are null-homotopic maps,
recovering the preceding definition (Stacks Project, Definition 22.26.1).
