What does the velocity dispersion of a galaxy mean? Also, what is the advantage of getting information about the velocity dispersion of a galaxy?
Definition of the velocity dispersion

From the title of your question, I'm unsure whether you actually know what "dispersion" means: The dispersion of some numbers is the spread around their mean, usually taken to mean their standard deviation. If you measure the velocity of several light sources (from the Doppler shift of their spectral lines) that are gravitationally bound, they will have some average velocity$^\dagger$. But since the light sources — which could be stars or gas clouds — move around each other, some move a little faster, and some move a little slower. This will cause their combined emission line to be broadened, instead of being a very narrow line. The central area of this line that emcompasses roughly 68% of the area of the line has a certain width. If the distribution is Gaussian (which is typically the case), then this width, divided by two, is the standard deviation. This is the usual definition, I think, but note that other definitions could be used, e.g. the full width at half maximum which differs from the standard deviation by a constant factor of $2\sqrt{2\ln2}$ for a Gaussian.

A relation between the mass, size, and velocity dispersion

The velocity dispersion $\sigma_V$ of gravitationally interacting "particles" such as stars in a galaxy, or galaxies in a cluster, provides a measure of the total mass and its distribution. The virial theorem, which describes the total energy of a bound system, can be expressed to show that the components of a galaxy of mass $M$ and radius $R$ will have characteristic velocities of the order of the velocity dispersion
$$
\sigma_V = \sqrt{\frac{GM}{CR}},
$$
where $G$ is the gravitational constant, and the observationally determined factor $C$ depends on the geometry and the actual mass distribution, as well as on whether the system is rotation-dominated (such as disk galaxies) or dispersion-dominated (such as elliptical galaxies) (Binney &amp; Tremaine 2008). For dispersion-dominated galaxies, $M$ refers to the total, dynamical mass of the system, and $C \simeq 6.7$ for various galactic mass distributions (Förster Schreiber et al. 2009). For rotation-dominated galaxies, the appropriate mass is the mass enclosed within $R$, and $C \simeq 2.25$, again averaged over various galactic mass distribution models (Epinat et al. 2009).

The velocity dispersion is rather easy to determine with spectroscopy, since it broadens the spectral lines (other processes also broaden the lines, so one must be careful to disentangle different effects). With photometry, the size of the galaxy can be determined (note though that the size can differ when observed in different bands). Thus, even if you don't know whether a galaxy is rotation- or dispersion-dominated, a moderately precise measure of its mass — one of the main characteristics of a galaxy — can be obtained (e.g. setting $C=5$ won't be too far off, and in astronomy we're usually happy with such small errors).

Other relations

Because of various scaling relations, $\sigma_V$ can also be used to probe other quantities, or other quantities can be used to probe $\sigma_V$. For instance, the M-sigma relation relates the mass $M_\mathrm{BH}$ of a galaxy's central, supermassive black hole to $\sigma_V$ (through $M_\mathrm{BH} \propto \sigma_V^4$, roughly), and the Faber–Jackson relation relates the luminosity $L$ of a galaxy to $\sigma_V$ (through $L\propto\sigma_V^4$, more roughly).



$^\dagger$In astronomy, this average velocity is called "the systemic velocity"), and is given by the expansion rate of the Universe (the Hubble law), plus in general a small velocity in a random direction (the "peculiar velocity" due to local dynamics).