The conceptor of the Amiga 1000 was Jay Miner,
who created the Atari 800 many years before. 
He wanted to make the most powerful computer ever, then he joined
a little California company called Amiga. 
He used the principle of the three coprocessors (again) 
to help the main processor. 

At the beginning, the Amiga had only 64 kilobytes of RAM!
The original "Amiga" called the Lorraine was meant to be a game machine
with some computer capabilities. 

Atari wanted to buy the Amiga but finally, Commodore succeded to buy it.
The Amiga 1000 concept and design predated any thought of the 
Atari ST machines. 
Tramiel bought the remains of Atari after his departure from Commodore. 
After the loss of a major legal battle for control of the Amiga chip set 
(Atari had bankrolled some of the chip set development) he and his 
sons came up with the Atari ST as a competitor for the upcoming Amiga. 

The operating system (AmigaDOS) was done by MetaComCo, 
a British company who specialized in the 68000 processor 
(they also made languages for the Sinclair QL). 
It is a fully multitasking system which looks like UNIX with a 
graphical user interface.
It was the very first personal computer with great graphics and sound 
capabilities with a GUI environment.
The Amiga BASIC was written by Microsoft (like most other versions of BASIC),
 but first models were shipped with a non-Microsoft BASIC called ABasiC. 

The Amiga 1000 will lose popularity one year later with the creation of
its two main successors: the Amiga 500 and the Amiga 2000. 

There was two versions of the Amiga 1000. 
The first one sold only in the USA, had a NTSC display and no EHB video mode.
Later versions would have this built in. The second one had a PAL display, 
the enhanced video modes (EHB) and was built in Germany. 

The official name for the A1000 was the Commodore Amiga. 
It was only when the A2000 was launched that they officially began to refer
to the machine by its model number. 

TRIVIAS:
The A1000 casing featured the signatures of all the designers and the 
paw print of Jay Miner's dog moulded into the inside of the casing.
When it was demonstrated at the CES show in 1984 it was just 4 breadboards
cabled together. They were so fragile each board had its own seat on the 
plane to Chicago. According to rumour it was going to include a phone 
answering machine built in, but Commodore decided it would be too 
expensive to fund.
Source : Amiga Interactive Guide 

the original Kickstart 1.0 disk was duplicated from a disk that the 
developers were using to note source code around. If you look on the 
ADF disk image past the first 256kb, you will see source files just 
sitting there, rather than with releases afterwards which were on 
previously-blank disks.
Source : Scott Lawrence. 

(www.old-computers.com)