Maybe FFT as used here means Fast Fourier Transform. That is a mathematical tool that can take a complex waveform, such as the sound of a baby crying, and break it down into a series of sine waves of different frequencies, amplitudes and phase angles with respect to the fundamental tone. When these sine waves are summed together again, the original tone is recreated. The sequence may have a very large number of terms, but when just the first few are summed together, a fairly accurate rendition of the original tone will be heard.
More specifically, an FFT is a transform on a discrete (sampled) sequence of any integer length, but best done on power of 2 lengths (although multi-radix lengths work well, too, unless the length is a prime number). It is a subset of the Discrete Fourier Transform that allows calculation to be done proportionally to n * log2(n) operations, as opposed to n*n operations (matrix multiply) in the DFT.
It is the sampled version of the Fourier Integral, but by being sampled, and thereby of limited length, it is an exact transform not only in the amplitude space, but also the power space, unlike the Fourier Integral, which is accurate in power terms (zero error on reconstruction by the inverse transform) but that has zero-energy errors (small spikes of zero width, hence zero power) in the amplitude domain at discontinuities in the original signal. This "error" is not germane to any kind of real signal, since no real signal can have energy at infinite frequency, and thereby has no actual discontinuity in the mathematical sense. "ears" due to bandwidth limiting are not this phenomenon.
The DFT/FFT, not having infinite length (Thank you Gauss for figuring this out), has no such phenomenon.
An FFT has exactly the same number of complex output terms as it has complex input terms.
Norman Morrison's book on "Fourier Analysis" is a very good way to learn this stuff. It is, however, <expletive> expensive.
Somewhat confusingly, the usual FFT on a real signal has one real output at DC, one real output at FS/2, and (n-2)/2 positive frequency complex terms, which are echoed by as many complex conjugate negative frequency terms. So, again, n samples in, n samples out is the case, but with all but two of the terms in the transformed data being complex, except at DC and FS/2. It is possible to have an "odd frequency" FFT that simply has n/2 complex terms, and the resulting conjugate terms.
There are other transforms, like the DCT,that are real to real transforms. They have more interaction between adjacent terms, but are also 1:1 and onto, or tight frames, or orthonormal, using your choice of term. One typically uses whatever is most useful in the application. In general, many transforms can have an FFT-like structure for computation, but no, not all of them. For grins, look up the 'KLT'. Have your favorite headache meds handy if you're stale on your linear algebra.
As a separate note, it is unwise to say "just the first few" are summed together. If you mean "the most energetic" you would be closer to right, but ordering in an FFT output is by frequency, so summing the first few terms via IFFT is a very uncomfortable, not very useful, form of bandwidth limiting.