What are baroclinic and barotropic waves?
Barotropic flows are those for which the fluid density $\rho$ depends on pressure $p$ only, i.e. $\rho = \rho(p)$. For example, constant density, isothermal, and isentropic flows are barotropic. Baroclinic flows are flows that are not barotropic. In this case the fluid density depends on pressure AND temperature, salinity, constituent concentration. Heavy fluid sitting adjacent to lighter fluid will sink and rise, respectively. 


  These flows are said to be baroclinic, meaning pressure breaking.


Consider the two fluid elements below. In the barotropic case, lines of constant $p$ are parallel to lines of constant $\rho$  so there is no mechanism to generate vorticity. On the other hand, in the baroclinic case, the lines of constant $p$ and $\rho$ are not parallel. The latter arrangement give rise to changes in vorticity $\vec{\omega}$ and circulation because the net pressure force no longer passes through the center of mass of the fluid element.



Seen another way, take a look at the vorticity equation with density variation:

$$\frac{D\vec{\omega}}{Dt} = (\vec{\omega}\cdot \nabla)\vec{u} + \frac{1}{\rho^2}(\nabla \rho \times \nabla p)$$

The cross term on the right hand side is the baroclinic torque. When it is zero the flow is barotropic whereas if it is nonzero the flow is baroclinic.

When applied to waves in the Ocean the result is that barotropic relates to waves propagating at the ocean surface. Baroclinic relates to waves propagating along density surfaces in the interior of the ocean (or internal waves).

One good example is the surface tide and the internal tide. As the surface tide (regular tide) sweeps around the main oceanic basins, it rubs against the ocean floor giving rise to internal tides. The former are called barotropic tides, whereas the latter are called baroclinic tides.



References:

P. Kundu, Fluid Mechanics

B. Sutherland, Internal Gravity waves