On June 27th, 1972, Bushnell and Dabney (who already had the Syzygy company that made Computer Space - so Computer Space if not technically an Atari product) incorporated as Atari.
They hired Alcorn, and the story goes that as mostly a training exercise for Alcorn, ended up with Pong. Bushnell supposedly took the name "Atari" from the game of GO, where it means a group of stones in direct threat of capture. Atari apparently has some other related meanings, such as getting something by luck, say winning the lottery. If you watch the tv show 12 Monkeys, one episode is called Atari, and a character uses the term as an adjective - being 'in atari' is being in a position where there are either no moves left or any move still ends in loss.
That original company only lasted about 4 years before Bushnell sold it to Warner for the financial means to bring the VCS into wide production and distribution. In 1984, Warner sold off much of Atari to Tramiel as Atari Corporation. That version of Atari released products like the ST, the Lynx, and the Jaguar. The company really stopped being a player when Tramiel sold it off in 1996. Since then "Atari" has been mostly about its logo and assets and who owns what.