Explain the internals of digital multiplexers
To understand how a multiplexer works, take a look at this picture:

You have \$n=2\$ address lines and \$2^n=4\$ data lines. What the decoder does is it selects only one of the data lines. What happens is you put your address in, and only one of the decoder outputs becomes high. Next the signals go through the AND gates. The selected AND gate has output \$I_k 1 = I_k\$. All the other AND gates have outputs \$I_m 0 = 0\$ since all other decoder outputs are low. Finally, the OR gate is used to combine the outputs of the AND gates.
Multiplexers are one of the basic building blocks in digital electronics. They are used in a verity of applications. Anywhere when you want to connect something with many outputs to a single input such as : busses, memory management units, ALU function select etc.