The distinctive badge symbolises the 1932 merger into Auto Union of Audi, Horch, Wanderer and DKW, each with distinguished histories of their own.
Well before that merger, however, it was August Horch who laid the platform for the Audi brand as we know it, following a dispute with former business partners that prevented him using his name on cars. A friend's son recognised the link between Horch, which means 'hark' or 'hear' in German, and the Latin translation of same, which is 'audi'. Horsch took the monicker and ran with it, the brand eventually becoming part of Auto Union.
In 1965 the company was renamed more simply Audi. It's now owned by Volkswagen.