OS/2 was supposed to be the next DOS, developed in co-operation between Microsoft and IBM. Microsoft dropped their end of the deal when Windows 3.0 gained popularity. For awhile, and because they'd shared APIs, IBM was able to run Windows apps within OS/2 Warp, but the compatibility plan wasn't really in their best interests.
IBM did go on to continue to improve on OS/2 with Warp coming out in 1994 -- before Windows 95 was out, adding pre-emptive multi-tasking, improved memory management, and a solid networking stack. But by then Microsoft's FUD was in effect, and most consumers waited for the next version of Windows.
OS/2 did hold a fair bit of ground in banking and in embedded systems (like ATMs) and although its no longer supported by IBM, a product called eCommstation continues its development and technologies.

Alas, it does not work in VMWare, as they long ago decided support wasn't worth it. Works fine in VirtualPC, however. This guy has some tips, and you can copy the floppy images right off the CD and use them (without conversion) in VirtualPC.
Once I gave up on VMWare (won't boot past the second floppy, no matter how I configured the VM) this was one of the easier virtual machines I've built. Video was decent at SVGA, sound worked pretty well (the installer likes making funky noises.) Still messing with networking.

Hit the jump for some screen shots...

The Floppy part of the install requires 3 disks that you can make (or copy) from the CD.

Once you're in the GUI, you can set-up your hardware.

The installer takes a few minutes, and makes random happy sounds while doing it.

The Warp Guide annoyingly tells you about every part of every dialog. You can turn this off!

The OS/2 Warp Menu. Look familiar, maybe?