Baroclinicity is a measure of the misalignment of the density of a fluid and its gradient of pressure. It is proportional to:
∇p × ∇ρ
i.e., the sine of the angle between the surfaces of constant pressure and constant density.
Baroclinic Instability is instability of an Atmosphere on a rotating planet due to baroclinicity. It is the source of much weather phenomena such as cyclones, and also is considered a means by which Protoplanetary Disks affect Planetary Migration.
A Baroclinic Atmosphere is an atmosphere where density depends upon both Temperature and pressure whereas a Barotropic Atmosphere is one where density depends only upon pressure.