It was Apple's version of DOS (also called ProDOS... not to be mistaken for MS-DOS, DOS just stands for Disk Operating System). If you need an OS for an Apple II, you can find some of the stuff you need here (including version 3.2 of the System Disk).
What set the Macintosh apart from the Apple II was things like a super fast processor (Motorola's 68000 running at 8 MHz), tons of RAM (128k, twice what the average home PC had) and much of the important aspects of the operating environment stored on ROM within the computer (this was how you could have the GUI based OS, some apps and files all on a 400k floppy).
The first (and I think only) Apple II to have a GUI based OS was the Apple IIGS... if you have one of those you'll need to go here for it's software (System 6.0.1). The interface is Mac-like, but it still isn't a Mac (nothing else is ).
SHowing my age, I had an Apple II in 1980. It used DOS (Apple DOS) and later used ProDOS. Those were the only official OS for that computer that I can remember. You could run CP/M, but as mentioned earlier, required a card to do so.
I had an Apple IIc when I was a kid. I used it to do my BASIC homework from computer class (which also had Apple IIe computers). The Apples in my school and at home ran Apple's DOS 3.3, and it came with Applesoft BASIC already on the diskette. ProDOS was also available and was a major improvement over DOS 3.3. This is not to be confused with IBM's or MS's DOS, which ran only on IBM PCs and clones. This was a Disk Operating System that really actually used Applesoft BASIC right off the bat.
The only Apple II machine that blew all of them away was the Apple IIGS, which was a fusion of Apple II and Macintosh technology. It could run the older AppleDOS versions (DOS and ProDOS), but also had a System Finder like the Macintosh. Unfortunately, it didn't catch on like it should have since most people only used the older Apple II software with it and a lot of the graphical apps ended up being developed for the Macintosh.