A probability density is just like any other kind of density; it's the amount of probability per unit volume. (In one-dimension, the 'volume' equivalent is length.) You need it when you have a continuous belief space but not when you have a discrete belief space. If you're doing billiard ball physics with point masses, you don't need mass densities; likewise if you're comparing billiard ball beliefs rather than real ones (the weatherman doesn't say "Rainy" or "Sunny" but expresses a percentage) you don't need probability densities.