If a distribution (here in 3D space) has a density $f(x,y,z)$, that means that for (almost) any volume in that space $V$, you can assign a probability to it by integrating the density (here "density" means probability per unit volume, very similar to, say, the concentration of salt in a solution):

$$P(V) = \int_V f(x,y,z)dV$$

From the above, you can see why it's also a "differential probability" (although I've never heard that term).