See the http://en.wikipedia.org/wiki/Discrete_Fourier_transform for the discrete fourier transform explanation. A very good book about implementing the Cooley Tuckey algorithm for FFT is Oran Brigham as cited there (if memory serves me right). There are also newer algorithms, like Winograd's. In Linux, there are dedicated packages that provide libraries for fast fourier transforms.

I believe the windowing applied to the data (Hanning, Hamming, cosine, etc) is one of the parameters taken by the FFT function call- for sure this is the case if you use Matlab and its DSP toolkit.  Otherwise, just implement it in your own loop before feeding the data to the C-T FFT algorithm.

You may try http://www.mathworks.com/matlabcentral/newsreader/view_thread/281031

Good luck

Thanks !!