Mainframe is a term used to describe very large, powerful, and expensive computers mainly used by large companies for bulk data processing (such as bank transaction processing).
The term arose during the early 1970's with the introduction of smaller computers such as the DEC PDP series, which became known as minicomputers, so users coined the term "mainframe" to describe larger, earlier types.
Mainframe computers abilities are not such much defined by their cpu speed as much as massive internal memory, large, high-capacity external storage, high-quality internal engineering and thus incredible reliability, and expensive but high-quality technical support. Mainframe computers can and do run successfully for years without interruption, with repairs taking place whilst they continue to run. Mainframe vendors offer such services as off-site redundancy — if a machine does break down, they offer the ability to run customers' applications on their own machines (often without users even noticing the change) whilst repairs go on. The internal redundancy of these computers can be such that, in at least one reported case, technicians could move one from one site to another by disassembling it piece by piece, and reassembling it at the new site, whilst leaving the machines running. The switchover took place entirely transparently.
Several manufacturers produced mainframe computers in the 1960's and 1970's, but today IBM produces the vast majority of them. Servers based on lower-cost microcomputer designs could be deployed at a fraction of the cost and offer local users much greater control of their own systems, and "dumb terminals" used for interacting with mainframe systems were gradually replaced by personal computers. Consequently, demand has plummeted and demand has been restricted mainly to financial institutions. However, with the development of large "web farms", IBM has found a new market for their mainframes, as they can offer similar web server performance as hundreds of smaller machines, but with much lower power and administration costs.
(According to a quick web search it looks like Hitachi, Fujitsu, and Amdahl are also still selling new mainframe computers. I rather suspect the Amdahl's are rebadged Fujitsus, but i'm not sure.)
See also supercomputer.