Derived functors are mathematical tools used in category theory and homological algebra to extend the concept of functors. They provide a way to measure how far a given functor is from being exact, which is crucial in understanding the relationships between different algebraic structures. Derived functors are often associated with concepts like cohomology and homology. The most common derived functors include left derived functors and right derived functors. These are constructed from a given functor by applying a projective or injective resolution, respectively. This process helps in analyzing properties of modules or other algebraic objects, revealing deeper insights into their structure and behavior.