AFAIK, Atari was started by Nolan Bushnell, but he eventually borrowed a lot of money from Time-Warner to help make the Atari 2600 as big as it was, so they got control over the company.
Bushnell was a nerd and a dirty hippie (oooo... I'm gonna get it for saying that ^_^), so he didn't get along with the suits at Time Warner (but will I get it for saying that?). After Bushnell said that the 2600 needed to be dropped by Atari, and replaced with something more modern and powerful and expensive, the suits thought he had lost his mind, so they had him replaced. Time Warner milked the 2600 until the videogame industry collapsed.
Then Time Warner split Atari into "Atari" (the hardware maker), and "Atari Games". And sold the hardware division to the Tramiel family.
Time Warner decided to completely change Atari's image, so they renamed them "Tengen" and started making NES games. Then they sent a lawyer into the patent office, saying they needed the specs for the NES's security chip, because Nintendo was suing them for breaking the lockout (which they weren't). Then using Nintendo's own blueprints, they broke Nintendo's lockout and made pirate games, and got sued by Nintendo. They said, "See, we told you so. Nintendo's suing us. Just like we said." Nobody fell for it.
The Tramiels tried to make the "Jaguar" console and the handheld "Lynx", but both were basically failures, and the Tramiel's hardware-making Atari went out of business. (Where are they now, I wonder?) I personally thought it would've been cool for Microsoft to buy the Atari-hardware rights before making the XBox. "Microsoft Presents: The Atari XBox!" But, they didn't. Ah well.
Atari Games eventually got picked up by Midway, who made that Wayne Gretzky hockey game for the N64, and probably a bunch of other stuff, but eventually gave up on sticking the "Atari" name onto things.
Infogrames bought what was left of Atari Games in order to get some of their rights and licences, and evenutally said "Hey, what they heck, we're allowed to do it." and changed their name to "Atari".