Often times, it refers to the degrees of freedom (df) in scaled-inverse-chi-square priors for variance parameters. This is one of the parameters that determines the degree of shrinking estimates towards the prior. Usually, such a df is chosen to be 4 or so, so that enough but not too many (i.e., too influential) moments exist.
As a side note, the inverse-chi-square distribution with 1 df is a very strange one -- if you take average of samples from this distribution, the variability of the average is even larger than that of any single observation....
