in very short and simple words, it is a signal transformation that represents the signal in different domain which is frequencey domain transform Fn(time) into Fn(f). why we need such transformation ? because there are some information are hidden in the time domain which is appear in the frequency domain. for example the MP3 and JPEG .. etc all this algorithms based on transform the time domain signal to frequency domain signal and making some process on the last one like deleting the uniimportant components and return it back to time domain to reduce its size without affecting its quality. 

Sandip,

I'll rephrase Yasser's (correct) answer.

Any signal can be represented as a time-based data set - such as a record of voltages made with a meter at regular intervals of time. Imagine logging a regular sinusoidal signal - it falls, rises, falls, and so on.

The space needed to record this infinite series of data would be very large.

But, one might notice that this long list of numbers is a representation of a single *frequency*  - rather than fill a notebook with lots of data, one could simply write, "Signal is a 10 volt peak-to-peak dataset of 0.01Hz frequency"

That one sentence captures all of the signal's information (except for phase...)

So, one often finds that taking the Fourier Transform (breaking a signal in time, into one of frequency) allows one to package information more densely - and to spot features that otherwise would not be visible.

I very much enjoy the responses on this topic on a different website, which is linked below. The first response gives a fun example in the context of planetary orbits (the referenced link is also included below).

http://math.stackexchange.com/questions/1002/fourier-transform-for-dummies

https://www.youtube.com/watch?v=QVuU2YCwHjw

I am thankful for all the answers. Does clear my doubts.

Regards 

The answers were focusing on time to frequency domain transformations (which is the basic definition anyway), that can be used in signal processing for audio-quality/noise cancellation, etc. But the basic mathematics is applicable to any domain transformations. So a closely related application is in image processing. Since a bmp (or tiff) image is essentially a sequence of values (between 0-black and 255-white; for 8 bit bmp) for each pixel, the bmp file data can be visualized as a 'value vs position' signal. Taking a simple example, if the image is a series of vertical black and white lines, its data when plotted vs pixel position will be a square wave, with width (frequency) corresponding to the thickness (in pixels) of the lines. When we do a Fourier transform (usually a fast Fourier transform (FFT) algorithm, which is optimized for digital signals), we can get the different harmonics, and can be used to filter out noise, or analyze the various aspects etc. This helps for e.g. in material sciences to analyze strain, fracture patterns, atomic lattice, etc with high-resolution images. Fourier Transform is used in spectroscopy, to analyze peaks, and troughs. Also it can mimic diffraction patterns in images of periodic structures, to analyze structural parameters. Similar principles apply to other 'transforms' such as Laplace transforms, Hartley transforms. These are chosen based on the type of the original signal (for e.g. periodic or not, discrete or continuous, etc.).

A major advantage of frequency domain is reducing typical computation times for convolving two signals. As pointed out already by @Abhilash Sugunan, Fourier transforms are so important in image processing. For such a 2D tasks we use the convolution theorem which states that: If f(x) and g(x) are two functions (in spatial domain) with corresponding Fourier transforms F(u) and G(u) (in frequency domain, then the Fourier transform of the convolution f(x)*g(x) is simply (becomes) just the product of the Fourier transforms of the two functions, F(u).G(u). This opens the door for use of the Fast Fourier Transform (FFT) for computing Discrete Fourier Transform (DFT) very efficiently. By using FFT, performing convolution in the frequency domain can be hundreds of times faster than conventional convolution.

Adding to @Mahmoud Omid's comment, no simple words can describe the importance of convolution theorem in engineering. It is a way of getting a mathematical expression of the output signal (of many real-life systems, e.g. filters, gratings, control systems, et al.) for any given input signal and system function. Fourier transforms and FFT, Laplace transforms, Z-transforms, etc are crucial for calculations in many engineering fields.

Dear Sandip,

You had got enough of wonderful answers and valuable suggestions that you must be fully satisfied. Hence I need not add any more.

All the best and regards

May I add one more point, which (as far as I could see) did not appear yet in the previous answers, but which is also a crucial application of the Fourier transformation? It lies in the field of differential equations, which are ubiquitous in theoretical physics.

In fact, if f(x) is an arbitrary function and g(k) its Fourier transform,

f(x) = \int dk g(k)*e^{ikx} ,

then the differentiation in "real space" corresponds to a multiplication with ik in "Fourier space". This means, the Fourier transform of the derivative f'(x) is given by ik*g(k), since

f'(x) = \int dk ik*g(k)*e^{ikx} .

This property of the Fourier transformation can be used to solve differential equations, because a complicated differential equation in real space may turn into an equivalent, but much simpler algebraic equation in Fourier space.

Typical applications in physics include the solution of the wave equation in classical electrodynamics [1, Chap. 6.4] or the free Schrödinger equation in quantum mechanics [2, Chap. II.12].

References: see, for example,

[1] J.D. Jackson, Classical Electrodynamics, 3rd ed., John Wiley & Sons, Inc., 1999.

[2] A. Messiah, Quantum Mechanics, vol. 1, North-Holland Publishing Company, 1961.