DOS 3.3 was the most common before ProDOS. How did it differ to DOS 3.2 and what versions were there before that?
There was no public release of DOS 1 or 2. DOS 3.1 was actually the first release to the public. It had a pretty significant bug in its MASTER CREATE program, and so the patched version DOS 3.2 was released. DOS 3.1 dates back to about June 1978.
The system master disk for DOS was able to operate on any machine regardless of how much memory it had. Simply running
INIT HELLO (or whatever) generated a disk that put RWTS (the DOS code itself) in exactly the position of memory it was now. A master disk would put it as high as it could on a 16, 32, or 48k machine. So a bug in the code that made new disks that could do that was a problem.
Anyway, DOS 3.1 and 3.2 used a 13 sector disk. But it wasn't too much longer before Woz realized that 16 sectors was possible with a minor hardware change to the Disk II controller card. DOS 3.3 came along with the necessary bits to fix that, and the DOS 3.3 System Master disk came with tools for moving files from 13 sector to 16 sector, and for booting the old disks from the System Master disk.
ProDOS came along later and worked completely differently. It more or less came onto the scene around the time of the Apple //e. It clearly was developed for and on the Apple ][+, but Apple did not release it until the Apple /// was pretty much completely obviously dead. Then suddenly here comes the //e and this new ProDOS thing. And Apple withdrew permission at that point for A2UGs to distribute DOS 3.3 and older anymore, intending to kill off the old stuff.
It never really did die off. ;)
There were other DOS releases made by Apple not explicitly referenced here. Wikipedia's Apple DOS page (linked above) notes that bugfix versions were released for all versions of DOS, without changing the version number. I do not doubt they did this because I have seen multiple builds of the same ProDOS version--though in the case of ProDOS, two versions may differ by as little as the build date on the splash screen.
There were also ProntoDOS, RDOS, and a whole bunch of mods to Apple's DOS, fundamentally all patches to the DOS RWTS to change the order things were written when the disk was INIT'd. You'll discover more about that if you dig into "DOS order" and "ProDOS order".
There were also patched versions of the RWTS to work with devices other than the Disk II such as UniDOS, which could work on a 400k 3.5" disk. The DOS RWTS could cope with a device up to 400k in size with the slightest patches, but it was only ever intended to work with the Disk II as Apple released it. ProDOS has Disk II code included (since the Disk II is about as dumb an interface as you can get), but it also has stuff to handle a "smart" block device. As far as ProDOS is concerned, all 3.5" disks and hard drives are smart block devices.
Hope that's a good start for you. If you're curious about the non-Apple DOS releases, others who know more about them can probably help you better. The only alternative DOS I've ever used was UniDOS, and at the time I used it, I didn't know how it was changed from DOS 3.3.
There were also two other formats common on the Apple ][: Pascal and CP/M. Though the latter required a special Z80 CP/M card (such as the Microsoft Softcard), the former was proprietary to the UCSD Pascal system that was released in the late 1970s.
I believe the idea was that since UCSD Pascal was a cross-platform p-Code environment, the format would be usable on other microcomputers at the time. However, I can't claim from personal experience ever seeing that in practice. As well, the only commercial software I recall using that was in the p-Code environment was Hayes Smartcom for the Micromodem ][.
As a side note, the Sider hard drive system for the Apple II series did support these two disk formats for partitions along with the standard DOS 3.3 and ProDOS volumes.