When Mac OS X was in development, Apple knew that what they were coding was a very different beast from the classic Mac OS. While Mac OS 9 was not a bad OS, per se, there were obvious limitations which betrayed its age. Its multi-tasking had been added in System 7, being tacked-on as MultiFinder in System 6. Its memory protection was woefully inadequate. Crashes were common. Something had to be done.
Rhapsody, the heavily NeXTSTEP-influenced OS which evolved into Mac OS X, was incompatible with old Mac OS applications. These applications could be ported to Rhapsody using the Carbon API, which was successfully used in the final Mac OS X, but however would it run non-'carbonised' legacy applications? The Mac had a software library spanning back over 15 years, and Apple could hardly toss that all aside.
The solution was what Apple termed the 'Blue Box', in contrast to the 'Yellow Box' which was Rhapsody's own native environment. The Blue Box could be launched on startup, on demand, or when a Mac OS application was launched, and would boot a copy of Mac OS 8 from within Rhapsody in which to run the program. Obviously, this required an installed copy of OS 8 to exist on the system, but if the user needed to use their legacy software, they could from within Rhapsody. One great advantage of this was that, as the Blue Box existed as a process within Rhapsody, it was protected from the rest of the system; if a program running within the Blue Box crashed, instead of taking the entire system with it, it would simply crash the Blue Box.
Rhapsody never shipped, beyond a few Developer Previews, but was instead absorbed into the Mac OS X project. Rhapsody became the code-name for Mac OS X Server 1.0, which was released with the same Platinum appearance theme as Mac OS 8 (and the Rhapsody Developer Previews). Mac OS X continued to be worked on, becoming Apple's next-generation consumer operating system. With Mac OS 9's software library continuing to grow, there was just as big a need for this software to be executable within OS X as before, if not greater. The Blue Box was polished and refined, and was revealed to the world as Classic.
Classic is the Blue Box, but updated to work with Mac OS 9. Or rather, Mac OS 9 is updated to work from within Classic; several key files are added to an OS 9 system folder in order to render it usable. (The process name for Classic reveals its roots, being called the TruBlueEnvironment.) As with Rhapsody, Classic can be asked to launch on log-in, can be manually launched (from either the Classic panel in System Preferences, or by running Classic.app) or will automatically launch when a legacy program is run.
As Classic starts up, it shows a progress bar in a window which disappears as soon as Mac OS 9 has booted. This window can be expanded by clicking the downwards arrow, revealing a scaled window showing the actual OS 9 boot sequence! Starting with the traditional Happy Mac, the boot image will appear (showing the Mac OS version number and the Mac OS smiling faces logo), and INITs will appear at the bottom as they would when booting Mac OS 9 itself. As soon as it has finished loading, this window will disappear and Classic will remove itself from the Dock. (You can keep its icon in the Dock if you wish by clicking and holding on it - you can then select "Keep in Dock" from the resultant menu). If Classic was invoked by running a program, that program will then appear on screen, and if launched from System Preferences can be asked to run without extensions, as if you were booting while holding Shift. The desktop can be rebuilt from here, too.
Programs running within Classic keep their Apple Platinum appearance. The menu bar at the top of the screen will also revert to Apple Platinum, seeing the return of the classic Apple menu and Application Menu which work in exactly the same way as they would under 'true' OS 9. They also appear in the Mac OS X Dock, and can be switched to from there. From the Application menu, you can switch to any currently running program — both those Classic and native OS X. (Switching to the Finder, alas, switches to the Mac OS X finder.) Running Apple System Profiler will show the system to be running whatever version of Mac OS 9 Classic is set to boot from, and will reveal the Gestalt ID and Model Name to be "Classic version x.x" and "Classic Mac OS Compatibility", respectively. (x.x will be the build number of Classic.)
Classic was intended to help the transition from Mac OS 9 to Mac OS X, and is slowly being phased out. Apple no longer ship any computers capable of booting from OS 9 and are removing Mac OS 9 System Folders from their new machines. The classic Mac OS's days are officially over, Steve Jobs having given it a funeral on-stage, and soon will be gone forever — Classic will no longer function on the new, Intel-based Macs which Apple will begin shipping from 2006, nor does it work in Mac OS X 10.5 (Leopard).