The fourier transform is a mathematical relationship between a function in the time domain, f(t), and one in the frequency domain, F(ω), and can be written as;
In the left panel of the figure below two sine waves with different amplitudes and periods are shown. Fourier transformation of these curves leads to the peaks in the right panel. The amplitudes of the sinusoids gives the size of the peaks after fourier transformation, while the period, or frequency, of the sine waves gives the offset of the peak from the zero frequency position in the transformed spectrum. Since the frequency of the blue signal is less than that of the red, the blue peak is closer to zero in the transformed spectrum than the red peak. The zero frequency position corresponds to the basic frequency of the spectrometer (600 MHz for 1H spectra on our spectrometers).
The peaks in the fourier transformed spectrum are infinitely thin because it is possible to measure exactly the frequency of the signals in the time domain. Unlike the example above, real nmr data decay with time in an approximately exponential manner. Applying the fourier transform to exponentially damped sinusoids results in lorentzian lineshapes. The width of the peaks is related to the rate at which the time domain signal decays; the more rapidly decaying the signal the broader the peak. One way to think of it is the signal decay makes it harder to determine the frequency and thus the peak position in the frequency domain is harder to define and the peaks broader.
The left panel of the figure below shows two exponentially decaying sine waves with different frequencies and decay rates. The blue sinusoid has a lower frequency and a faster decay rate than the red. As in the figure above, the lower frequency of the blue signal means that the peak obtained after fourier transformation is closer to the zero frequency position than the red signal which has a higher frequency. The decay of the signals means that the peaks are no longer infinitely thin; they now have some width. The faster decay rate of the blue signal compared to the red translates into the blue peak being broader than the red after fourier transformation.
One might note that the figures above are starting to look like real NMR data. This is because an experimental FID is simply the sum of many decaying sinusoids. The fourier transform has the fortunate property that the fourier transform of multiple frequencies is simply the sum of the fourier transform of each of them. This makes it possible to easily convert from the time to frequency domain by picking apart the signal one frequency at a time.
In both of the figures here I have arbitrarily given the blue peak a negative offset from the zero frequency and the red peak a positive one for the sake of simplicity and clarity. Determining whether a peak's offset is positive or negative requires either using two detectors, or splitting the signal in two and phase shifting one half by 90o. The latter method is now generally used but whichever method is applied the result is two sets of data with a phase difference of 90o between them. These sets correspond to the real and imaginary components of the complex number in the equation at the top of this post, and after fourier transformation they correspond to the real and imaginary spectra that enable phase correction, a subject for another post.