Amiga Corp. was founded in 1982 by Dave Morse, Jay Miner and Larry Kaplan, and first called 'Hi-Toro' (which was quickly changed because a lawnmower-company had a similar name); so they decided to call it Amiga, spanish for 'girlfriend'. Some of the Amiga's developers, namely Dale Luck, R.J. Mical and Jay Miner, wanted to build a professional computer - but the investors wanted a videogame, since the videogame-market was booming these days. The compromise was a videogame-system concept that could easily be expanded to a computer (which was ok for the investors as long as the changes didn't cost too much). So finally, the first prototype, 'Lorraine' was created, more a patchwork of wirewrapped VLSI prototype boards than a final design. When the videogame-market then crashed in 1983, only small modifications were necessary to have a professional computer, the Amiga 1000 (just called 'The Amiga' in the beginning). In most A1000, by the way, the developers have left their signatures, imprinted in the plastic case.
The Amiga truly was a revolution: it was the first homecomputer ever to have preemptive multitasking, 4096 colors to be displayed at once (EGA-equipped PCs had 16 colors back then) and stereo-sound, controllable by an efficient and fast operating system, AmigaDOS. It was no surprise that several companies were interested in Amiga Corp., companies like Sony, Hewlett-Packard, Philips and Apple. Jack Tramiel (just being fired from his own company, Commodore), bought ATARI these days, which had financial problems due to the losses in the videogame-market. He wanted revenge (yes, that's real life!), and so tried to buy Amiga Corp. before Commodore could. But his opponent succeeded in buying Amiga, and everything turned out to be good (at first).
The real impact started on Consumer Electronics Show (CES) 1987 in Las Vegas, where Commodore introduced new machines for mass- and professional markets, Amiga 500 and Amiga 2000. They were best-sellers, and Commodore led the homecomputer market again, for many years (the A2000, with its open system architecture, was also quite successful in professional markets). Then followed, among others, the revolutionary but ill-fated CDTV and, introducing AmigaOS 2 and the 'enhanced chipset' (ECS), A3000/ A3000T, A500 Plus and A600. By that time, Amiga sales slowly began to break in, mainly because Commodore management had missed many opportunities to improve their machines' market shares (such as an offer from Sun Microsystems to market the A3000UX as a Unix-workstation); additionally, new developments were held back because of the cost. The 'Advanced Amiga' chipset, for example, was introduced with A1200 and A4000 in 1992, although working prototypes (A3000+) already existed much earlier. And the CD³², which could have been a breakthough in the videogame markets, appeared shortly before the end in 1993, when Commodore was already unable to satisfy the demand. Finally, after Commodore was liquidated in 1994, all Amiga rights and patents were sold to other companies. Some great concepts, such as the AAA-chipset (being modular and with up-to-date graphics and sound), never made it into production, although in 1993 some AAA prototype-boards (called 'Nyx') already existed.

Looking back from today, the Amiga wasn't just a computer, and the Amiga developers weren't just developers. They were idealists, perfectionists - ingenius minds that set their goal to building the most innovative (and creative) computer in history. And they did. Sadly, their management often worked against them, so Commodore went bankrupt in the end and took the great ideas into its grave.

Amiga was always a passion more than a job.
Dave Haynie, from "The Deathbed Vigil"

Today, a new AMIGA Incorporated holds most of the rights, but it is unlikely that new hardware will be produced. They're working on some new AmigaOS-projects for different platforms. See the Amiga website for actual developments.