A multiplexer is really very simple at its basic level. The most basic mux has three inputs and one output. One input acts as a switch to control which of the other two inputs is sent to the output. So say if the switch is 0 then the output=input 0, and if the switch is 1 output=input 1. This is a 2:1 mux. You can then scale this up to an nxm sized mux and even add other signal processing to make it more advanced.
Like /u/Dynamic_Physics says, it‘a a switch to choose one of multiple inputs/sources for a signal.
The common reason computers (laptops) have it is to directly connect the screen to either the graphics chip built into the processor to save power or a fast gaming graphics card for performance.
This can be done without a MUX switch by always sending the signal through the slower graphics to the screen even if the gaming GPU is producing that picture. But that can decrease performance.
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