Question: **What are those characteristics by which every sound can identified uniquely?**

For example, pitch is one of the characteristics of sound, but let’s say a note C# can also be played on a guitar and piano with same pitch but the resulting *sound* that we hear is different so what are those characteristics which defines every sound.

Answer: 
For a long time, timbre was believed to be based on the relative amplitudes of the harmonics. This is a hypothesis originally put forward by Helmholtz in the 19th century based on experiments using extremely primitive lab equipment. E.g., he used Helmholtz resonators to "hear out" the harmonics of various sounds. In reality, the relative amplitudes of the harmonics is only one of several factors that contribute to timbre, and it's far from sufficient on its own, as you can tell when you listen to a cheap synthesizer. Flicking the switch from "flute" to "violin" doesn't actually make the synthesizer sound like a flute or a violin enough that you could tell what it was intended to be.

A lot of different factors contribute to timbre:

* relative amplitudes of the harmonics
* the manner in which the harmonics start up during the attack of the note, with some coming up sooner than others (important for trumpet tones)
* slight deviations from mathematical perfection in the pattern $f$, $2f$, $3f$, ... of the harmonics (important for piano tones)
* the sustain and decay of the note (guitar versus violin)
* vibrato

Some sounds, such as gongs and most percussion, aren't periodic waveforms, in which case you don't even get harmonics that are near-integer multiples of a fundamental.

Because there are so many different factors that combine to determine timbre, it's remarkably difficult to synthesize realistic timbres from scratch. Modern digital instruments meant to sound like acoustic instruments often use brute-force recording and playback. For example, digital pianos these days just play back tones recorded digitally from an acoustic piano.
Very few sounds you hear are regular vibrations at a single frequency (i.e. *monochromatic* sounds). Rather, most sound-producing objects will cause vibrations in the air at many different frequencies.

Mathematically, one can apply the [Fourier transform](http://en.wikipedia.org/wiki/Fourier_transform) to the function that describes the vibration over time. The result is a function that describes how strong each frequency component is.

Many instruments are decently approximated as consisting of a *fundamental* pitch and a discrete sequence of *overtones* on top of that, where the frequencies of the overtones are nice-looking rational (often integral) multiples of the frequency of the fundamental. Which overtones are present and what their relative strengths are comprise the primary components of the [*timbre*](http://en.wikipedia.org/wiki/Timbre) (characteristic) of the sound. A C# will have the same fundamental frequency of $466\ \mathrm{Hz}$ or so on both a guitar and a piano, but the strengths of the overtones (such as at $932\ \mathrm{Hz}$ or $1398\ \mathrm{Hz}$) will be different.

By the way, if you have a broad continuum of frequencies present at approximately equal strengths, the result is known as *noise* (a technical term, yes). There are certain standard types or [*colors*](http://en.wikipedia.org/wiki/Colors_of_noise) of noise depending on the specifics of the spectrum of frequencies.