A supercomputer is the fastest computer you can get. They are really expensive and are only really used for specialized applications that require the huge amount of computing power the supercomputer can give out. For example weather forecasts, animated graphics, fluid dynamic calculations, nuclear energy research, and petroleum exploration. Like I stated in the mainframe section supercomputers uses all there power into a single program rather than the mainframe does many programs at the same time but it is slower this way. They also look the same as a mainframe.
Multiprocessing is where a computer is able to processes multiple programs at once, UNIX is one of the most used multiprocessing systems, there are others like OS/2 for high end personal computers. Multiprocessing systems are more complicated than single process systems because the system needs to allocate resources to completing processes in a suitable way.