Commodore Business Machines (CBM) was founded by Jack Tramiel in 1958 and located in Silicon Valley. At first they produced calculators and business machines in general. But in October 1976 Tramiel bought MOS Technology. Its founder, Chuck Peddle, who had invented the famous MOS 6502-microprocessor in 1975, was eager to build a whole microcomputer around his 6502, and so convinced Tramiel to build the PET ('personal electronic transactor'). It was first presented to public at the famous West Coast Computer Faire, 1977 (where Apple also introduced the waypointing Apple ][) and became a great success. That was the birth of Commodore's microcomputer series, intended to take a piece of the big cake called 'microcomputer market', which Apple had more or less invented.
Many 8 bit-machines machines followed (all based on the MOS 6502 and its derivates), like the CBM/PET 2000-, 3000-, 4000- and 8000-series (for the professional markets), or the VIC-20 (Codename MicroPET) for the consumer market; there were even a chess-computer (the rare CHESSmate) and two Pong-clones (TVG2000K and TVG3000H). But Commodore's greatest success was undoubtly the Commodore 64 (Codename 'VIC-30'), the 'best-selling computer ever'. It made Commodore leader in the homecomputer market for nearly a decade.
Commodore had also built many creative and bizarre prototypes, like the CBM 900 (a machine with the rare Zilog Z8001-CPU, running 'Coherent', a Un*x-derivate), or the C64DX (also known as 'C65', with 4096 colors, two SID-chips, built-in 3.5" drive and BASIC V10, see picture below), which was developed to inherit the C64's market position (following the interesting 264-series and the C128, who had both failed on that). These and other prototypes might have changed Commodore's future for the better (then maybe not), but sadly they were mothballed and never produced in quantities.

The C65, however, was cancelled for another reason: Commodore succeeded in overtaking AMIGA Inc. (a struggle with their opponent ATARI), and the C65/ C64DX project became obsolete. Buying AMIGA was an ingenius decision (maybe the last ever made) - AMIGA technology simply was a masterpiece and waypointing in every respect, especially multimedia. Additionally, just for the sake of market presence, Commodore created a complete PC line to take a piece of the 'IBM compatibles' cake (see advert above). This PC line rooted in the PC-5/ PC-10 introduced in 1984 (both based on the i8088), and included other interesting machines like the PC-1 (stripped down and very compact PC-10), PC-30/ PC-40 (i80286), COLT-series (name for the PC-line outside of Europe) and others (up to i80486). Maybe the PC line was Commodore's biggest mistake, a lesson that other famous homecomputer companies (like ATARI and AMSTRAD) had to learn as well: with their unique designs and special solutions (such as the Advanced Graphic Adaptor, AGA, or self-designed mainboards), Commodore PCs could never compete to the masses of cheap clones. As usual, most people didn't care about quality, just price.
What's worse, Commodore was never good at marketing: the PET-series and the C64 had almost sold themselves, so management seemed to believe it to be unnecessary. Both AMIGA and PC line sales suffered from that. So finally, after a period of stagnation, Commodore was liquidated in 1992 (also see the AMIGA section for Commodore's last days). A sad story, if you ask me - Commodore products were of a high quality, and often incorporated interesting and clever solutions.