After programmer John Carmack and three of his fellow Softdisk employees -- John Romero, Adrian Carmack (no relation to John), and Tom Hall -- developed a Super Mario Bros 3 mockup around a Platform Game engine he'd created (view gameplay of Dangerous Dave in Copyright Infringement here), the group struck out on their own and founded id Software. They were major supporters of the Shareware distribution model, and it paid off; at one point in the early '90s, the first episode of Doom was on more hard drives than Microsoft Windows.
John Carmack is the only one of the original founders who remains with the company, and has earned a reputation as one of the industry's most brilliant programmers. As a result of his "programming first" mentality -- he once commented notoriously that stories in video games are like stories in porn; they're expected to be there, but aren't important -- many id releases are Tech Demo Games that show off the features in a new engine, which they then license out to other developers.
It's worth noting that, whenever a new IdTech engine is built, the previous one is released under the GPL licence, thus continuing id's legacy in Warsow, Nexuiz, Alien Arena, Urban Terror (previously a Quake III Arena Game Mod) and OpenArena, among others.
Currently, id Software is part of Zenimax Multimedia (who also houses developers Bethesda Game Studios, creators of the Elder Scrolls saga, Tango Gameworks, and others). In 2011, id announced that it would no longer be licensing out it's idTech engines (though still releasing previous engines). Instead, Carmack will focus solely on in-house development (which he loves) engine development for id and the greater Zenimax studios.
For the record, it's pronounced as one syllable (ID, with a short I, not EYE-DEE), after the part of the Freudian psyche that deals with instinct and pleasure-seeking. In other words, perfect for a company making games.
Among the games id Software has developed:
- Commander Keen series
- Wolfenstein series
- Doom series
- Quake series