Supercomputer is a computer that led the world (or was close to doing so) in terms of processing capacity, particularly the speed of calculation, at the time of its
introduction. The term “Super Computing” was first used by the New York World newspaper in 1929 to refer to large custom-built tabulators IBM made for Columbia
University. It was used to solve problems involving by major universities, military agencies and scientific research laboratories.
Additional Source: http://en.wikipedia.org/wiki/Supercomputer
Mainframes are designed to handle very high volume input and output (I/O) and emphasize throughout computing. It is used mainly, by large organizations for critical applications, typically bulk data processing, industry and consumer statistics, ERP, and financial transaction processing. The term probably originated from the early mainframes, as they were housed in enormous, roomsized metal boxes or frames.  Later the term was used to distinguish high-end commercial machines from less powerful units which were often contained in smaller
packages. Today in practice, the term usually refers to computers compatible with the IBM System/360 line, first introduced in 1965. (IBM System z9 is IBM’s latest incarnation.) Otherwise, systems with similar functionality but not based on the IBM System/360 are referred to as “servers.” However, “server” and “mainframe” are different (see client-server).
Additional source: http://en.wikipedia.org/wiki/Mainframe_computer
Minicomputer is a midsized computer. It is an old term for a class of multi-user computers, lies between the largest multi-user systems (mainframe computers) and the
smallest single-user systems (microcomputers or personal computers). In the past, it formed a different group with its own hardware and operating systems.
Additional source: http://en.wikipedia.org/wiki/Minicomputer