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An application is a term used in computing to define a program or group of programs that perform a function. Business application software is a package (can be custom built but packaged as a single unit) that carries out a business function such as accounting or payroll.
Early software performed very narrow functions. It was not unusual to have a program do a single calculation on all of the data records, which were often cards or tapes, and then have the same data run through another program to do another calculation. This is because computers had very little memory and could not handle multiple actions at once. Today's systems group programs into "Applications," that manage a full business function.
Software applications can be stand-alone modules or part of a larger system such as an ERP, which integrates multiple applications sharing one set of data. Stand-alone applications are generally easier to implement and start using, but add difficulty when you want to share information or report from an organization management point of view because data will need to be gathered and transformed to a single format.