AmigaOS is the native operating system of the Amiga personal computer, a series of workstations originally marketed by Commodore from 1985 through 1994, and later by other companies. For a brief history of AmigaOS and the Amiga in general, see Wikipedia's Amiga history page.
AmigaOS is notable as being an early GUI operating system, following in the footsteps of systems such as Mac OS and GEM (although GEM was only considered part of the OS on Atari computers -- see Wikipedia's Atari TOS page for more on that topic). Unlike Mac OS, and like most other GUI systems, AmigaOS also included a command line interface.
The command line portion of the OS, implemented by Kickstart, is called AmigaDOS. While AmigaDOS is not to be confused with PC-style DOS, it does share some similarities with DOS and other text-mode systems.
The actual GUI is called Workbench, and as the name suggests, instead of the now-common GUI metaphor of a desktop (with files and folders), the screen uses the metaphor of a workbench, with tools, gadgets, drawers, etc. (Originally, "Workbench" was the name of the entire OS; version 3.1 was the first version to be called "AmigaOS".)
A modern operating system that seeks to replicate AmigaOS is AROS (AROS Research Operating System, formerly Amiga Research Operating System). AROS seeks to be "compatible with AmigaOS at the API level", meaning that programs written for AmigaOS should work under AROS by just recompiling.
Another OS that is mostly compatible with AmigaOS is MorphOS. Unlike AROS, MorphOS is mostly binary-compatible with AmigaOS when run on actual Amigas (meaning that many Amiga programs will run under MorphOS without change).