Potential density
Potential density is a dynamically important property: for static stability potential density must decrease upward. If it doesn't, a fluid parcel displaced upward finds itself lighter than its neighbors, and continues to move upward; similarly, a fluid parcel displaced downward would be heavier than its neighbors. This is true even if the density of the fluid decreases upward. In stable conditions (potential density decreasing upward) motion along surfaces of constant potential density (isopycnals) is energetically favored over flow across these surfaces (diapycnal flow), so most of the motion within a 3-D geophysical fluid takes place along these 2-D surfaces.
Barotropic fluid
The density of most liquids is nearly constant (isopycnic), so it can be stated that their densities vary only weakly with pressure and temperature. Water, which varies only a few percent with temperature and salinity, may be approximated as barotropic. In general, air is not barotropic, as it is a function of temperature and pressure; but, under certain circumstances, the barotropic assumption can be useful.