The Fourier transform is the limit as L->oo of the ComplexFourierSeries.
Put more colloquially, take a periodic, smooth, continuous function (these conditions can be relaxed). It can be thought of as the accumulation of lots of sine waves of different frequencies and amplitudes, some of which are phase shifted.
The Fourier Transform of your function is another function that tells you the amplitude (and phase) at each frequency.
Specific example: The Fourier Transform of a sine wave is a spike function. There is only one frequency that has a non-zero amplitude.
Specific point: If your original function really is periodic then only frequencies that are integer multiples of the original period can occur. These are the harmonics, and the Fourier Transform tells you the relative intensities of them. Please note that there are details concerning multiples of 2*pi in this that I am ignoring.
One other point. FFT is an algorithm for computing a finite version of the FT. Please avoid using FFT as "short-hand" for "Fourier Transform" as some people, rightly or wrongly, will interpret it as a sign of ignorance.
See also FastFourierTransform