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: 
The entirety of the wave train is involved in the perception of [timbre](http://en.wikipedia.org/wiki/Timbre) and it is not reducable to a few easily measurable features. There are entire books, including one by Helmholtz, investigating the characteristics of sound and how it relates to perception. I like *Music, Physics and Engineering* by H. Olson.

Often the power spectrum of a given sound is used to help analyze the timbre of notes; the relative magnitudes of the different harmonics, and the degree to which their frequencies differ from the theoretical ideal are a good starting point. However, these Fourier transform based approaches are not complete in that they typically do not address the perceptually salient features such as note attack and decay.

Another approach to take is to examine how synthesizers generate sounds; for this I recommend checking out the Sound on Sound article series [Synth Secrets](http://www.soundonsound.com/sos/allsynthsecrets.htm).
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.