+1 for the Brigham reference. It's the best explanation I've ever read.
@andand: thanks, yes, excellent book, even though it's quite old now: the applications chapters are good too, and still relevant.
The FFT is just an efficient implementation of the DFT. The results should be identical for both, but in general the FFT will be much faster. Make sure you understand how the DFT works first, since it is much simpler and much easier to grasp.
When you understand the DFT then move on to the FFT. Note that although the general principle is the same, there are many different implementations and variations of the FFT, e.g. decimation-in-time v decimation-in-frequency, radix 2 v other radices and mixed radix, complex-to-complex v real-to-complex, etc.