Volume number density is the number of specified objects per unit volume:
where N is the total number of objects in a volume V. 
Here it is assumed that N is large enough that rounding of the count to the nearest integer does not introduce much of an error, however V is chosen to be small enough that the resulting n does not depend much on the size or shape of the volume V because of large-scale features.
Area number density is the number of specified objects per unit area, A:
Similarly, linear number density is the number of specified objects per unit length, L:
Column number density is a kind of areal density, the number or count of a substance per unit area, obtained integrating volumetric number density along a vertical path:
It's related to column mass density, with the volumetric number density replaced by the volume mass density.