Mainframe systems or simply mainframes are computers used for critical application, bulk processing and transaction processing.
They are larger than the basic computers we find around. They are advantageous as they are able to process large data faster. For this reason, large institutions like banks tend to use them for processing of data because of the thousands of transactions they engage in.
They are capable of handling and processing very large amount of data quickly.
Examples of other large institutions that uses mainframe computers apart from Bank stated in the question are
Real time system means that the system is subjected to real time, i.e., response should be guaranteed within a specified timing constraint or system should meet the specified deadline. For example: flight control system, real time monitors etc.
real-time system has been described as one which "controls an environment by receiving data, processing them, and returning the results sufficiently quickly to affect the environment at that time".The term "real-time" is also used in simulation to mean that the simulation's clock runs at the same speed as a real clock, and in process control and enterprise systems to mean "without significant delay".
