The Full Width Half Maximum (as defined by John Rennie's answer ) can have the following physical meanings:
- For a Lorentzian lineshape spectrum, it is proportional to the square magnitude of the strength of the coupling between electromagnetic field and the atomic transition begetting the spectrum in the first place;
- For a Lorentzian lineshape spectrum, it is the reciprocal of the $1/e$ lifetime $\tau$ of the atomic transition, which, in the Lorentzian lineshape case, has the unique memoryless lifetime probability density function, namely $p(t) = \exp(-t/\tau)/\tau$;
- For spectrums dominated by Doppler broadening, say in thermalized gas molecules the FWHM is proportional to both the square root of temperature, but it is also proportional to the centre frequency.
For more understanding of the points (1) and (2), see the section "The Shape of the Spectrum Without a Cavity" in my answer to "Why do lasers require mirror at the ends?" and the model of the excited atom coupled to the ground state electromagnetic field there, as well as the alternative and equivalent Wigner-Weisskopf theory - this is referenced in my answer too although by far the best exposition I know of (Weisskopf's / Wigner's own) is in German, there is no direct English language translation and the main English exposition I know of is:
Section 6.3, M. O. Scully and M. S. Zubairy, "Quantum Optics"
which is not the grandest piece of technical writing I know. Basically it works like this: the atom is coupled roughly equally to all modes of the electromagnetic field (at least, the coupling is roughly flat near the transition frequency), so it tries to couple equally to all. However, only those modes that are tuned to the transition can be coupled into successfully; for the others, the beating between the EM field mode and the atomic transition thwarts the radiation into that mode by destructive quantum interference. The stronger the coupling constant near the transition frequency, however, the more this detuning and destructive interference effect can be overcome and thus the broader the spectral line, which, in this theoretical model, is indeed exactly Lorentzian:
$$H(\omega) = \frac{1}{(\omega - \omega_0)^2 \tau^2 + 1}$$
where $2/\tau$ is the angular frequency FWHM,
$$\tau = \frac{1}{2\,\pi\,|\kappa|^2}$$
is the transition lifetime, $\kappa$ is the (complex) coupling co-efficient between each electromagnetic field mode and the transition and $\omega_0$ is the transition frequency. The probability amplitude to find the atom still excited a time $t$ after observing that it is excited is:
$$\psi(t) = \left\{\begin{array}{ll}0;&t<0\\\frac{1}{\sqrt{\tau}} \exp
\left(-i\,\omega_0\,t - \frac{t}{2\,\tau}\right); & t\geq0\end{array}\right.$$
A common reason for a nonthermalised line to deviate from Lorentzian lineshape is that it is owing to a sequence of transitions rather than a lone transition. In that case, the spontaneously radiating atom is not memoryless. See my answer to "Are there old aged particles?".
For thermalized gas transitions, the spectrum will be broadened in a way related to the Maxwell-Boltzmann velocity distribution. See the Wikipedia page for "Doppler Broadening".
