Left derived functors are a concept in homological algebra that extend the idea of functors to measure how far a given functor is from being exact. They are constructed using projective resolutions of modules, allowing mathematicians to study properties of modules and their relationships through derived categories. The most common left derived functors are the Tor functors, which help in understanding the behavior of tensor products. These functors provide insights into the structure of modules over a ring, revealing information about their extensions and cohomological dimensions.