Intuition was the widget and graphics system that made the whole thing work. It was not implemented primarily as an application-managed graphics library (as most systems, following Xerox' design, have done), but rather as a separate task that maintained the state of all the standard UI elements independently from the application. This made it singularly responsive because UI gadgets were live even when the application was busy. The Intuition task was driven by user events through the mouse, keyboard, and other input devices, and it also arbitrated collisions of mousepointer and icons, controlled the "animated icons" etc.
Due to a mistake made by the Commodore sales department, the first floppies of AmigaOS which were released with Amiga 1000 named the whole operating system (OS) "Workbench". Since then, users and CBM itself referred to "Workbench" as the nickname for the whole AmigaOS (including Amiga DOS, Extras, etc.). This common consent ended with release of version 2.0 of AmigaOS, which re-introduced proper names to the installation floppies of AmigaDOS, Workbench, Extras, etc.).
Workbench is also used on the Amiga as a metaphor for their own standard of "desktop" as opposed to others, such as "Macintosh Finder". Workbench itself is another library or process. Rumors said that this concept of modularity was invented by Commodore to treat Workbench as a window amongst the others in the desktop, in order to avoid reprisal from Apple. But this can only be considered a rumor, as all patents on windowed GUIs were property of Xerox at that time.
Early versions of AmigaOS did treat the Workbench as just another window on top of a blank screen; but this is due to the ability of AmigaOS to have invisible screens with a chromakey or a genlock -- one of the best features of Amiga platform -- even without losing the visibility of Workbench itself. In later AmigaOS versions Workbench could be set as a borderless desktop.
Amiga users were also able to boot their computer into a command line interface (CLI), also known as shell. This was a keyboard-based environment without the Workbench GUI. Later they could invoke it with the CLI/SHELL command
LoadWB which performs the task to load Workbench GUI.
Like most GUIs of the day, Amiga's Intuition followed Xerox's lead anteceding solutions, but pragmatically, a command line interface was also included and it extended dramatically the functionality of the platform. Later releases added more improvements, like support for high-color Workbenchs screens and 3D aspect. Often Amiga users preferred alternative interfaces to standard Workbench, such as Directory Opus, or ScalOS interface. An interesting article about these replacements is available here (in French language).
The use of improved, third party GUI engines became common amongst users who preferred more attractive interfaces -- such as Magic User Interface (MUI), and ReAction. These Object Oriented graphic engines driven by "classes" of graphic objects and functions were then standardized into the Amiga environment and changed Amiga Workbench to a complete and modern guided interface, with new standard gadgets, animated buttons, true 24bit-color icons, increased use of wallpapers for screens and windows, alpha channel, transparencies and shadows as any modern GUI requires.
Unfortunately, MUI and similar systems abandoned the separation of the User Interface between the application (which specified gadgets to be displayed) and the Intuition task (which actually managed gadgets for all applications, even when they were busy), so much of the responsiveness of the original system was lost.
Heirs of Workbench are nowadays: Ambient for MorphOS, ScalOS, Workbench for AmigaOS 4 and Wanderer for AROS. There is a brief article on ambient and descriptions of MUI icons, menus and gadgets here (aps.fr) and images of Zune stay at main AROS site
A new object-oriented toolkit for all Amiga-like platforms (AmigaOS, MorphOS, AROS), Feelin,was introduced in 2005, and makes extensive use of XML guidelines. It uses its own memory management system, and its memory-pools system shares the embedded OS's semaphores. Feelin also features a non-centralized ID allocation system, a crash-free object invocation mechanism, and an advanced logging system. Details and images on Feelin can be found at its website