Geos started as a graphical environment for the Commodore 64, 128 and the Apple II microcomputers.
8 bit versions
The 8 bit versions of GEOS was a single tasking, graphical operating environment. GEOS was best known at the time for bringing high quality fonts to low end printers.
There is a wealth of information available here.
GEOS programs were expected to be written in assembly, and with a massive memory constraint, there wasn't that many available.
From what I've seen, GEOS was available on both the Commodore 64 and the Apple II. Both machines were 6502 based, allowing a lot of code to be shared between the two platforms. I don't know if the same executables could run on the different machines, but I would expect them to be able to...
GEOS was very popular on the Commodore 64, and 128. It really took off when Commodore started bundling it with new systems.
While the desktop publishing revolution was happening on the MAC platform, GEOS had brought these capabilities, including great looking fonts to the lower class 8 bit machines. On the Commodore side, they even produced a mouse, the 1351 giving a real GUI on a very limited machine.
Berkley software also produced their own REU's (RAM Expansion Units) for the Commodore 64/128 allowing larger programs to run with a bank switching scheme, and even providing a ram disk, greatly speeding up operations.
Version 2.0 introduced limited color support, along with more device drivers for input devices, printers, and things like hard disks, and ram/cpu expansion.
16 bit versions
Soon it was obvious that the 8bit market was dying, and it was time for Geos to transition to new platforms. A new system was written in assembler that used the Motif look & feel to give the IBM PC a new GUI. AOL was an early adopter of Geos, and its early dialer programs used Geos.
For a small amount of time Geos was bundled with lower end systems, as it had a low system requirements. However their delays in giving out a developer SDK doomed the platform, as they were unable to compete with Microsoft Windows 3.0 & their onslaught.