What is the role of Multiplexer and Decoder?

A multiplexer (MUX) is a selector. It has many data inputs, a few select lines and one output. The select lines decide which one input is passed to the output. A 4-to-1 MUX has 4 inputs and 2 select lines. It is used to choose one source among many, for example in a bus system.

A decoder does the opposite job. It takes an n-bit binary code and activates exactly one of its $2^n$ output lines. A 3-to-8 decoder has 3 inputs and 8 outputs. It is used to select one register, one memory chip or one device from an address.