What is a Buffer and why do we need it

When i first entered into hardware field i always used to listen to the word called buffer. what it exactly mean i was thinking.
So if you give a signal at the input of a buffer the output will be same. but why do we need that kind of circuit.( It isnt wastse of money i thought).

When ever we want to transfer a signal from one stage of any circuit to other stage we use buffers but why?Essentially a buffer should have very high input impedance and low output impedance why?Example:

Suppose i am getting a signal from a circuit with impedance of say 400 ohm, now if i connect it to a buffer( essentially a buffer will have high input impedance).

Hence the voltage at the input of the amplifier becomes

V= Vsignal * (Rbuffer/Rbuffer+400)

since Rbuffer is very high( input impedance of buffer) hence the signal which is coming to the input stage of buffer will not vary.

now the second thing is the output of buffer is almost the same as the original signal.Hence a buffer amplifier essentially gives the same output voltage what it gets at its input but it gives a flexibility in such a way that, since the ouptut impedance of the buffer is low it can be used to drive more circuitary and also there wont be any extra loading on the input side of the buffer.

hence we are essentially increasing the fanout of the buffer.