CUI and GUI are two techniques used to send commands to a computer processor. Both accomplish the same goal, but employ somewhat different methods to do so. CUI, invented by Xerox in the seventies, predates GUI, which was introduced by Apple in 1984.
CUI stands for character user interface and can also be referred to as the command prompt, C prompt or command line. In the direct operating system called DOS this is generally displayed on the screen by a "C:>" or "C>". This is the point at which commands are written out before being sent to the computer by pressing "Enter" on the keyboard.
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GUI stands for graphical user interface and eliminates the need to type commands on the C prompt or command line. With a GUI, you use a mouse to move the cursor around the screen, place it on an icon and click. By doing so a command is sent that would otherwise have to be typed using the CUI.