Atari is a brand name used by many companies since its inception in 1972.
Atari's first incarnation was primarily responsible for the formation of the modern video game industry. They developed the Atari VCS (later known as the Atari 2600), which was one of the first game consoles and one of the highest-selling consoles of all time (with 30 million units sold). Despite this, the video game crash of 1983 would cause Atari's first incarnation to go bankrupt. Nintendo would cite the crash and Atari's lack of quality control for the 2600 as a reason for their strict third party licensing policies throughout the 80s and 90s.
As Atari, Inc., Atari ported Donkey Kong, Donkey Kong Jr., and Mario Bros. to the Atari 2600. Atari, Inc. was also in talks with Nintendo to license the distribution rights to the Family Computer, but the deal fell apart.
Atari, Inc. was split up in 1984 after going bankrupt, and the arcade and game software division became Atari Games Corporation. Atari Games would publish games for the NES using a subsidiary named Tengen Inc.; only three of Tengen's games were licensed by Nintendo (R.B.I. Baseball, Gauntlet, and Pac-Man). A series of legal battles with Nintendo stemming from the latter's strict licensing policies would result in Atari being barred from distributing NES games by a Federal judge. Atari Games was acquired by Time Warner Interactive in 1993.
What remained of Atari Games was eventually bought by Infogrames, a French video game company, in 2001. The two merged with other groups and ended up with the name of Atari, SA. Atari then focused more on publishing games and investing in the hardware side of the industry. After getting really close to bankruptcy, the company was revived with the help of aids from the American and French governments. Since then, Atari announced a new console called the Atari VCS in 2017, which is set to be released towards the end of the year.