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Computer programmers historically used "Press any key to continue" (or a similar text) as a prompt to the user when it was necessary to pause processing. The system would resume after the user pressed any keyboard button.
Early computers were typically operated using mechanical teleprinters, which provided a continuous printed record of their output. However, during the 1970s, these became obsolete and were replaced with visual display units, and text was lost once it scrolled off the top of the screen. To compensate, programs typically paused operation after displaying one screen of data, so that the user could observe the results and then press a key to move to the next screen.
These prompts were commonplace on text-based operating systems prior to the development of graphical user interfaces, which typically included scrollbars to enable the user to view more than one screen/window of data. They are therefore no longer required as a means of paginating output, but the graphical equivalent (such as "Click OK to continue") is still used for hardware interactions.
The prompt is not strictly accurate in that, for the vast majority of computer systems, modifier keys or lock keys would not cause processing to resume, as they do not produce an actual character that the program could detect.
Some Samsung remote controls for DVD players, as is the case of DVD-R130, have included an "anykey" to their interface. It is used to view the status of the DVD being watched. 
There are reports that some users have searched for such a key labelled "any", and called technical support when they have been unable to find it. The computer company Compaq even edited their FAQ to explain that the "any" key does not exist, and at one point considered replacing the command "Press any key" with "Press return key".
The concept of the "any key" has become a popular piece of computer-related humor, in part because of an episode of The Simpsons in which main character Homer Simpson asks "Where's the any key?" when confronted with the "press any key" prompt. Plastic "any keys" with adhesive backings are available as novelty gifts.