Its when you add vocal or aural effects to the sound of the sax, for example, the use of flutter tongue or the growl, to "dirty up" the sound. Rock, Rock and Roll and the Blues are where I feel a dirty sound comes into its own. Rasping, screaming solos and note bending all help to dirty up your environment! Pure filth. Bring it on!
A good example for me, would be Bobby Keys playing on Brown Sugar by The Rolling Stones. In my opinion, Keys uses a combination of both of these effects to very good use. He manages to use the flutter tongue and the growl at the same time as well as bending some of the sustained notes hat he plays.