Volume weighting is a technique for accommodating the Malmquist bias (and I think may be able to deal with the Eddington bias). The weighting is actually done using the inverse of the volume, thus it is sometimes referred to as inverse volume weighting.
The problem is dealing with systematic errors in the sampling of different luminosities (absolute magnitudes) in surveys. This method of dealing with it is to calculate a volume for each luminosity, consisting of the volume of space in which an object of that luminosity would be detected by the survey. Vmax is the term used for such a volume, i.e., a Vmax exists for each possible luminosity. When counting objects of a given luminosity, when using it statistically, multiply the count times 1/Vmax (and the technique is also known as the 1/Vmax method). Examples of applicability would be in deriving (from the survey data) a mean luminosity of objects within some a volume of space, or a density function of luminosities over such a volume.