What is meant by "correlation" when referring to spectral coherence I've been reading that coherence measures the correlation between two waves as a function of frequency. I also read that difference in phase does not mean less coherence at a given frequency, and that amplitude is also not a factor. Some explanations say that its is a measure of the shift in phase. As I understand it, a shift in phase requires a shift in frequency, but coherence is the measure of correlation at a given frequency. So I guess what I want to know is this:

If I've taken the Fourier transform of two waves, and am looking to see how they are correlated in the frequency domain, what am I actually measuring?

EDIT: I'm a biologist, and my knowledge of digital signal processing is pretty basic. I'm trying to pick this stuff up on the fly. So intuitive explanations, if possible, would be greatly appreciated.
The coherence function, as used in signal processing, measures the normalized correlation between to power spectra:

$$
C_{xy}(f) = \frac{|G_{xy}(f)|^2}{G_{xx}(f) G_{yy}(f)}
$$

where $G_{xx}$ is the power spectral density (PSD) of the signal $x(t)$, $G_{yy}$ is the PSD of $y(t)$, and $G_{xy}$ is the cross-spectral density (CSD) of $x(t)$ and $y(t)$.

The CSD is computed as the Fourier transform of the cross correlation of $x$ and $y$:

$$
R_{xy}(\tau) = \operatorname{E} \left[ x(t) y(t + \tau) \right]
$$
$$
G_{xy}(f) = \mathcal{F} \left[ R_{xy} (\tau) \right]
$$

You can think of the CSD as measuring "hidden periodicities" in the cross correlation, just like the PSD measures periodicities in the autocorrelation. (This is where the name for the "periodogram" PSD estimator comes from.)

Incidentally, if the cross-correlation has harmonic content at some frequency $f$, you can think of the signals being correlated at that frequency.