There is no written evidence that indicates who invented the lever and pulley. The earliest mention of a pulley-like device (a device which also used a lever) came from the Greeks. Archimedes, a Greek mathematician, was credited with discovering how levers work and developing pulley systems.
Archimedes invented the compound pulley, a device which consisted of multiple pulleys. His invention allowed the Greeks to move and lift extremely large, heavy objects without using too much manpower. According to Plutarch, Archimedes was able to move an entire ship (complete with cargo and passengers) using his compound pulleys. It is also said that due to his advanced understanding of levers, Archimedes uttered the phrase "Give me a place to stand on, and I will move the Earth."