The term multi-tasking is a little vague, and is used to describe different things. All modern OS's are multi-tasking, even phone OS's. Otherwise the complexities of receiving a call while playing a game would be huge.
I'm not particularly an expert on the matter, but this is how I understand it...
Mult-tasking from a users perspective: when you swipe from one app to the next, you are bringing one app to the foreground and sending others to the background. Depending on Settings > General > Application behavior, the OS will either continue to run the apps in parallel or "pause" the one that's in the background to conserve battery or improve performance of the foreground app. Despite being paused an app may still receive a little bit of time to do housekeeping tasks, but the OS can enforce a limit on the amount of time it has.
Multiprocessing is the term for describing the fact that the OS and all running apps are "spread" over the two CPU cores that the PlayBook has. It's handled automatically by the OS. It's perfectly possible to run the OS and multiple apps simultaneously on just one core, but having more cores gives us twice the processing power without consuming twice the battery power. It does make it a little harder for the OS to manage the cores. It's certanly not so that one core is for the OS while the other is for apps. Both cores are used to their maximum potential if needed.
The OS is constantly running multiple processes, even when you're not using your device. Each process can have different priorities and scheduling rules. Some of those processes may be apps, others may be services that the OS provides such as listening for incoming calls, checking the battery or sharing files over Wi-fi. Any process, including apps, can run multiple threads, which in some operating systems also have their own rules. So even an app can do multiple things at the same time, e.g. animate the screen and decompress the next picture.
QNX, the PlayBook's OS, is a microkernel-based Real-Time Operating System or RTOS.
RTOS means that the method used to switch back and forth between tasks is predictable, dependable and fast enough for it to be used in situations that require real-time precision, such as engine management of a car, or temperature monitoring of a nuclear power plant. On non-RTOS operating systems like Windows, there is no hard guarantee that a process will be running at a very specific time and able to manage a critical event such as the firing of a spark plug. If there is something else going on that prevents the process from running (paging to disk, for instance) it will just have to wait. For a nuclear power plant, or the brakes of your car, that may not be a good idea. How important all this is for a tablet is debatable, but it probably makes the whole multiprocessing circus a lot smoother & more efficient.
The microkernel model, as I understand it, is a way of seperating tasks, processes, threads, drivers and so on at the lowest possible functional level, and means that the OS can manage the schedules easily and recover from crashes better than monolithic kernel OS's. It's pretty complicated.
A lot more info on the PlayBook's "multitasking" can be read here:
The QNX Neutrino Microkernel
As for the competition, Android and IOS are not microkernel or RTOS, as far as I know, but are also multitasking operating systems. However, on IOS the developer has very limited access to those features. Apple policy dictates that it may only be used for specific functions, such as playing music, and that regular apps are paused or even terminated when the user switches to another app. They claim it is "better for the users" or something.