The existence of three different companies called 'Atari' causes untold confusion, not least because they were once peas in the same pod; indeed, they were both the pod and the peas at the same time, except that there wasn't a plurality of peas, there was only one pea, a big one, and that pea was also the pod. It was like an orange, in that it had its own pod, pea-peel.
In brief, Nolan Bushnell founded a company called 'Atari Inc' in 1972. In the wake of Pong's substantial success Atari Inc was bought by what was Warner Communications, a company which has subsequently undergone nomenclatural gymnastics of its own. Warned provided a lot of the cash which paid for the development and production of the Atar 2600 video games console, which was launched in 1978. Although the console was only a modest hit, the launch of the Atari 2600 'port' of arcade hit Space Invaders in January 1980 caused the 2600's popularity to skyrocket.
This Atari thrived. In 1980 it produced one-third of Warner's total revenue, four hundred and fifteen million dollars in 1980 money (up from $3.2 million in 1973). The company produced a clutch of legendary arcade games, consoles, and home computers, all under the Atari name, although each product range was developed by separate divisons of the company.
By 1982 Atari's 2600 console had had its day, whilst sales of the company's home computers were fading fast. 1982 was a bad year and 1983 was worse. The video games crash of that year caused Warner to sell the consumer electronics and home computer division to a man called Jack Tramiel, whose Tramiel Technologies named their acquisition 'Atari Corporation'. This company produced the ST range of computers, the 7800, the Lynx, the Jaguar and so forth, a separate and equally tragic history told elsewhere.
Warner continued to own the arcade division, calling it 'Atari Games Corporation', or Atari Games for short. The company was eventually allowed to become an independent venture, with Warner as a majority shareholder, at least until 1993, when the newly-reconstituted Time Warner bought Atari Games back again and renamed it 'Time Warner Interactive'.
Williams purchased Time Warner Interactive in 1996 and changed the name back to 'Atari Games', the company now a division of Midway, part of the William / Bally / Midway continuum. After the release of San Francisco Rush 2049 in 1999, Midway renamed the company 'Midway Games West'.
In 2001 Midway left the arcade market to concentrate on computer and console games. As Hasbro then held the rights to use the 'other' Atari for the same purpose this caused something of a quandry, and in 2003 Midway Games West closed shop, and that was that. Today Atari Games no longer exists, either in name or in spirit. Only in memory.
Now there is only one Atari, albeit that it is not actually Atari, indeed it is just a logo and a label, rather like Hello Kitty, but awe-inspiring rather than cute.
Frankie say 'no more'.
Selected sources
http://www.movieprop.com/videogames/atari/
http://www.gamespy.com/top10/february03/atari/
http://www.faqs.org/faqs/atari-8-bit/faq/section-84.html