Hydrostatic equilibrium (hydrostatic balance) in a body of fluid (e.g., gas) is the state such that forces cancel and the fluid remains still. A calm lake is at hydrostatic equilibrium. In astrophysics, a gas ball (e.g., star) or a planetary atmosphere is at hydrostatic equilibrium if at each level, gravity (toward the center) matches the upward pressure, i.e., for a spherical symmetric star:
dP Mrρ —— = -G ——— dr r²
Other forces (inertia, if the body is spinning, magnetic force, a nearby object's gravity) can be additional factors. A protoplanetary disk can take a flared torus shape in hydrostatic equilibrium due to the star's heating and radiation pressure.
An example of something not at hydrostatic equilibrium could be a gas cloud in the midst of collapse or expansion.
For many purposes, such as modeling stars, the state of a volume of material can be sufficiently close to hydrostatic equilibrium that it can be assumed for simplicity's sake. In other words, movement is trivially slow compared to the aspects under consideration.
A differentiated object is an astronomical object that has settled so that inner material is not uniform. A sufficiently massive object in hydrostatic equilibrium often forms layers (concentric shells) based upon density of the material. By some definitions, the distinction between planet and planetesimal is that the former is a differentiated object and the latter is not.