Hasbro was founded in 1923 by Henry and Merrill Hassenfeld (it was then known as Hassenfeld Brothers). The company originally sold textile remnants. It soon moved into manufacturing pencils and other school supplies; the first toy manufactured by the company was Mr. Potato Head in the late 1940s.
Hasbro became a dominant player in the game industry by acquiring companies like Parker Brothers in 1968 and Milton Bradley in 1984. Here's a more detailed look at Hasbro's game-related corporate acquisitions.