Derived functors are a concept in homological algebra that extend the idea of functors to capture more information about the relationships between categories. They are constructed from a given functor by applying a process that involves projective or injective resolutions, allowing mathematicians to study properties like exactness and cohomology. These functors help in understanding how certain algebraic structures behave under various transformations. For example, the derived functor of the Hom functor, known as Ext, measures the extent to which a module fails to be projective. This provides deeper insights into the structure of modules and their relationships in category theory.