In its most generic use the density is the mass per unit volume of an object or region and might have units like kg/m3 or Mo/pc3.
The Number density is the number of a particular object or species per unit volume and might be used when describing the number of electrons per cubic centimetre in a plasma or the number of stars in the core of a globular cluster.
Density profile is the density as a function of some variable like distance from the centre of a star.
The mean or average density is the total mass of a region divided by the total volume of that region.
Since the behaviour of matter is often a function of density it is a very important quantity to determine when trying to explain the underlying physics of an object.