Multiplexers are relatively simple devices. Essentially, they are an electronic switch of sorts, where a combination of switch inputs determine which data input to output. Multiplexers can come in many sizes, although the most common are the 2 and 3 bit versions.
Why are multiplexers important? Because they allow you to build logic circuits without using logic gates. The switch inputs (A,B,C,etc.) to a multiplexer are digits of a binary number, which selects the data input to output. This may say sound confusing, but let's look at an example of how a multiplexer is implemented.
Assume you need to construct a logic circuit that has the function F=A'B'C' + A'BC + ABC. Instead of using three NOT gates, three AND gates, and an OR gate, you can use a 3 or 2 bit multiplexer. To implement the circuit, you would connect A, B, and C to the switch inputs on the multiplexer. Next, you would ground all data inputs, except for the ones that are in the equation, which are connected to Vcc.
Alternatively, you can use a 2-bit multiplexer, as shown on the right in the diagram. In this case, you use input C instead of Vcc and connect it in accordance with the logic function (C' goes with data input A'B', C goes with data input A'B, etc.). This allows you to use a smaller multiplexer IC, but in some cases, such as the one shown below, you will also have to use a NOT gate.
Depending on your application, multiplexers can greatly reduce the complexity of your digital logic circuit. If your logic function is small, it may be wiser to use logic gates instead.