What's the significance of fast fourier transform and discrete fourier transform?? Is it used widely in practice?

Fourier transforms represent a standard, most well-known method to transform a signal (such as a price time series) from time domain into the frequency domain. DFT and FFT are just special cases for particular types of signals. DCT, which is another variant, is used, for example, by the JPEG compression algorithm. Google can help further.

Yes it's widely used in practice, specially for computer implementations. Like series with functions, discret fourier transforms provide a quick way to reach convergence for series approximations on derivatives pricing. Basically, fourier transforms provide a way to transform time into frequency for signal analysis. In finance, it's a way to transform a partial differential equation into an ordinary diffrential equation with stochastic calculus to price very quickly derivatives.

Yes it is. A lot of exotic derivatives need extensive calculus/time to be priced. Fast fourier transforms is a mathematical tool to reduce it by improving speeds of convergence.

The FFT is an algorithm that implements the DFT in an efficient manner by taking advantage of cyclic nature of the calculations (which are based upon sines/cosines). You may also want to do a search on Wavelets as well as they can perform a similar function.