It's a very complicated history. NOTHING AT ALL LIKE THE "FAKE" JAPANESE LUXURY BRANDS!
The company was formed as Auto Union AG by a merger of 4 smaller companies in 1932 - Audi, DKW, Horch, and Wanderer. Horch was the oldest, founded in 1899. The four rings in the Audi logo symbolize the 4 original companies. In 1948 Auto Union was dissolved by the Soviet military administration. The following year Auto Union GmbH was formed using Marshall Plan funding in part and began selling old DKW designs. In 1958 Auto Union GmbH became a wholly-owned subsidiary of the Daimler Group. In 1964 Volkswagen took over Auto Union GmbH. In 1969, NSU Motorenwerke AG, which had just been taken over by VW, and Auto Union GmbH merged to form Audi NSU Auto Union AG. In 1985, Audi NSU Auto Union AG was renamed as AUDI AG and remains so today.
cliff notes:
1932 - Audi, DKW, Horch, and Wanderer merge to form Auto Union AG
1948 - Auto Union AG dissolved by Soviet occupiers
1949 - Auto Union GmbH re-established
1958 - Daimler Group buys Auto Union GmbH
1964 - Volkswagen Group buys Auto Union GmbH from Daimler
1969 - VW buys NSU and merges NSU with Auto Union to become Audi NSU Auto Union AG
1985 - Audi NSU Auto Union AG renamed AUDI AG (all caps is proper name, BTW)