( Not done yet. It's a lotta work to convert Wikipedia paste-up into Stack Exchange paste-up. BTW, this text in the wikipedia article was done by me, probably over a decade ago. anyone is welcome to edit this to convert it.)
Group delay is a useful measure of time distortion, and is calculated by differentiating, with respect to frequency, the phase response of the device under test (DUT): the group delay is a measure of the slope of the phase response at any given frequency. Variations in group delay cause signal distortion, just as deviations from linear phase cause distortion.
In linear time-invariant (LTI) system theory, control theory, and in digital or analog signal processing, the relationship between the input signal, $x(t)$, to output signal, $y(t)$, of an LTI system is governed by a convolution operation:
$$y(t) = (h*x)(t) \ \triangleq \ \int_{-\infty}^{\infty} x(u) h(t-u) \, \mathrm{d}u $$
Or, in the frequency domain,
$$ Y(s) = H(s) X(s) \, $$
where
$$ X(s) = \mathscr{L} \Big\{ x(t) \Big\} \ \triangleq \ \int_{-\infty}^{\infty} x(t) e^{-st}\, \mathrm{d}t $$
$$ Y(s) = \mathscr{L} \Big\{ x(t) \Big\} \ \triangleq \ \int_{-\infty}^{\infty} y(t) e^{-st}\, \mathrm{d}t $$
and
$$ H(s) = \mathscr{L} \Big\{ x(t) \Big\} \ \triangleq \ \int_{-\infty}^{\infty} h(t) e^{-st}\, \mathrm{d}t $$
Here $h(t)$ is the time-domain impulse response of the LTI system and $X(s)$, $Y(s)$, $H(s)$, are the Laplace transforms of the input $x(t)$, output $y(t)$, and impulse response $h(t)$, respectively. $H(s)$ is called the transfer function of the LTI system and, like the impulse response $h(t)$, fully defines the input-output characteristics of the LTI system.
Suppose that such a system is driven by a quasi-sinusoidal signal, that is a sinusoid having an amplitude envelope $a(t)>0$ that is slowly changing relative to the frequency $\omega$ of the sinusoid. Mathematically, this means that the quasi-sinusoidal driving signal has the form
$$x(t) = a(t) \cos(\omega t + \theta)$$
and the slowly changing amplitude envelope $a(t)$ means that
$$ \left| \frac{d}{dt} \log \big( a(t) \big) \right| \ll \omega \ .$$
Then the output of such an LTI system is very well approximated as
$$ y(t) = \big| H(i \omega) \big| \ a(t - \tau_g) \cos \big( \omega (t - \tau_\phi) + \theta \big) \; .$$
Here $\tau_g$ and $\tau_\phi$, the group delay and phase delay respectively, are given by the expressions below (and potentially are functions of the angular frequency $ \omega$). The sinusoid, as indicated by the zero crossings, is delayed in time by phase delay, $\tau_\phi$. The envelope of the sinusoid is delayed in time by the group delay, $\tau_g$.
In a linear phas system (with non-inverting gain), both $\tau_g$ and $\tau_\phi$ are constant (i.e. independent of $\omega$) and equal, and their common value equals the overall delay of the system; and the unwrapped phase shift of the system (namely $-\omega \tau_\phi$) is negative, with magnitude increasing linearly with frequency $\omega$.
More generally, it can be shown that for an LTI system with transfer function \displaystyle H(s) driven by a [[phasor|complex sinusoid]] of unit amplitude,
: x(t) = e^{i \omega t} \
the output is
: \begin{align}
y(t) & = H(i \omega) \ e^{i \omega t} \ \\
& = \left( \big| H(i \omega) \big| e^{i \phi(\omega)} \right) \ e^{i \omega t} \ \\
& = \big| H(i \omega) \big| \ e^{i \left(\omega t + \phi(\omega) \right)} \ \\
\end{align} \
where the phase shift \displaystyle \phi is
: \phi(\omega) \ \stackrel{\mathrm{def}}{=}\ \arg \left{ H(i \omega) \right} ;.
Additionally, it can be shown that the group delay, \displaystyle \tau_g, and phase delay, \displaystyle \tau_\phi, are frequency-dependent, and they can be computed from the [[phase unwrapping|properly unwrapped]] phase shift \displaystyle \phi by
: \tau_g(\omega) = - \frac{d \phi(\omega)}{d \omega} \
: \tau_\phi(\omega) = - \frac{\phi(\omega)}{\omega} \ .