Darby was born in Coalbrookdale, Shropshire to Abraham and Mary (née Sergeant).
The Coalbrookdale Company also played an important role in using iron to replace the more expensive brass for cylinders for Thomas Newcomen's steam engines.
He and his partners were responsible for a very important innovation in introducing the use of coke pig iron as the feedstock for finery forges.
This formed a significant part of the output of Horsehay and Ketley Furnaces, which they built in the late 1750s.
He had married twice: firstly Margaret Smith (died 1740), with whom he had three children including Hannah who married Richard Reynolds, and secondly the Quaker minister Abiah Maude, with whom he had a further thirteen children although only four including Abraham Darby III survived.