"Mainframe" is an unscientific term that means different things to different people, but generally speaking, a mainframe is a multiuser business or scientific computer the architecture of which is optimized towards I/O (input and output).
If CPU power is brain and I/O is brawn, compared to a PC, a mainframe's brawn-to-brain ratio is tilted towards "brawn". Of course, the CPU of a mainframe is more powerful than a PC CPU, but a mainframe's I/O capabilities are even more impressive compared with those of a PC.
As high-entry economic proposition for a business to consider, a mainframe exists primarily to allow immense numbers of clients to interact with virtually infinite local and nearby storage and virtually infinite network connectivity. A mainframe can replace a Linux server farm. Do you need a Linux server farm in a box, one with faster machine-to-machine communication because each server is really a virtual machine under the VM operating system running across a cluster? Then you need a mainframe.