Why are star systems flat but planets are spherical? This video vaguely explains why the solar system (or other star systems and galaxies) are flat. 
So, does the same argument apply to planets? 
The general answer is that for a given volume, a sphere has the smallest surface area. But I fail to see why it isn't the case for galaxies. 
 A: The shape is determined by minimising the energy of the system for a fixed value of angular momentum.
The energy terms obviously include the (negative) gravitational potential energy, the rotational and internal kinetic energy of the constituents (the material of the planet or the gas and stars of a galaxy).
In the absence of significant rotation, then the energy is minimised in a symmetric spherical scenario. This is the case for planets. Any attempt to collapse along an axis will decrease the potential energy, but at the expense of raising the internal energy of the gas/fluid/material that makes up the planet by even more. Having said that, planets like Jupiter do rotate fast enough to be distinctly non spherical.
You might think that a similar argument would apply to a galaxy. However, what happens is that the increasing internal energy of the gas (NB it is important to note that the flattening occurred before most of the stars formed) in the proto-galaxy could be radiated away as photons. I.e. The gas gets hotter and radiates. This allows the collapse to occur, but a disk results in order to conserve angular momentum.
Now you could say, well why doesn't the same thing happen to a planet? The answer is to do with the relationship between internal energy of the planet material and it's density. Rocky planets are made of relatively incompressible material where there is a huge increase in internal energy (pressure) if they are squeezed. Gas giants are more compressible and more like a galaxy, but their centres are governed by electron degeneracy pressure. Such objects cannot radiate away much of their internal energy and if this dominates the rotational energy, they remain nearly spherical and any collapse is halted.
A: When a planetary system forms from a rotating cloud of gas and debris, the gravitational force acts in all 3 dimensions, but tbe centrifugal force of rotation offsets the gravity only in 2 dimensions of the plane of rotation. So over time the size in the third dimension (the thickness of the rotating disk) shrinks under gravity making the planetary system flat.
In contrasst, when a planet forms, gravity is offset by pressure in all 3 dimensions. Any deviation from the spherical form creates a pressure disbalance, so over time the shape tends to end up roughly spherical depending on the size and material. For example, in asteroids, the gravity is insufficient to affect the shape of the material, so they are usually of a random shape. Planets made of a gas or liquid would be more uniformely spherical with no "miuntains" on the surface.
Of course the rotation of the planet creates a centrifugal force stretching the shape of the planet at the equator from ideally spherical.
A: Well, that also applies to planets to some extent. Earth is bulged little on the equator due to the rotational motion. So, it is rotation that causes the systems to be flat.
Gravity is associated with "minimum surface area for a given volume". i.e. spherical shape
Exactly opposite is "maximum surface area for a given volume", which is flat/disc/saucer shape. Actually a sheet/plane but it is not practical to contain stuff inside a sheet, so, next close is a disc.
balanced/Opposed/shielded gravity are associated with this opposite shape which includes - solar systems, galaxies, and UFOs :).
The shape seems to provide a hint why UFOs are (appropriately) depicted as saucer shaped.
