The Omnic Crisis
Before the organization we now know as Overwatch was formed, there was the Omnic Crisis. 30 years before present day in the game, the humans had indulged in the creation of robots, also known as Omnics, and used them for personal gain. However, the Omnics began to develop a mind of their own and they rebelled against the humans, thus causing the Crisis. Unable to suppress the rebellion on their own, the United Nations called for the formation of the Overwatch.
Founded by Gabriel Reyes and Jack Morrison, the members of the organization fought back against the Omnics and essentially beat them back into submission. As a result, the humans began to feel disdain for the Omnics in general. There was also a religious group that formed after the Crisis, known as the Shambali, and it consisted of people who believed that the Omnics had souls.