The company name was taken from its founder Adolf (Adi) Dassler whose first name was shortened to the nickname Adi. Together with the first three letters of his surname it formed ADIDAS.
Adolf Dassler lived in the German town of Herzogenaurach with his brother Rudolph Dassler. In between the two World Wars they built a successful business making shoes specifically for sports called Gebruder Dassler.
The brothers fell out because they couldn't agree. Adi was a craftsman who loved nothing better than the peace of his workshop, whilst his brother was a more gregarious party loving salesman. They should have complimented each other, but sadly did not and went their separate ways - Adolf built up the ADIDAS business and Rudolph built the PUMA business.