Why are all planetary systems and most galaxies disk-shaped?

The shapes of large astronomical objects reflect three natural laws. First, the force of gravity from a faraway object decreases with the square of the distance — i.e., 2 times as far, so (½)² is ¼ the force — but does not depend on direction. Second, nature keeps things spinning. Spin can come from collisions or when galaxies are forming. Once an object gains some rotation, it retains this “angular momentum.” The only way to get rid of this momentum is to give the spin to some other object. Third, energy is conserved. The gravitational force meets this requirement because orbiting objects move faster near a mass and slower as they are farther out — the energy of motion converts into energy fighting against gravity.

The force of gravity makes some objects, such as the Sun, spherical. Every point on the surface of a sphere is the same distance from the center and thus at constant gravitational force.
 
Suppose a star rotates rapidly. The radial force of gravity isn’t the only influence pulling toward the center of the system; spin also affects it. Along the equator, rapid rotation reduces the pull of gravity and the object is more extended — it becomes a flattened sphere. This effect shapes Jupiter; its diameter at the equator is larger than between its poles.

Finally, consider a collapsing spinning gas cloud. Gas will need to keep its angular momentum constant, so the rate of rotation increases as it collapses under gravity. When the spin becomes close to the speed of a circular orbit, increases in rotational speed force the gas to move outward and get stuck in circular orbits. The originally somewhat-spherical gas cloud collapses with all of its material rotating in the same direction and in the same plane; spin plus gravity equals a rotating disk with a vertical thickness much less than its radius.
 
This process is widespread. Clouds that form stars make spinning circumstellar disks where planets are built; gas funneling into the centers of galaxies produces accretion disks around supermassive black holes; and the gas that makes galaxies gives us the thin, beautiful disks of spirals. — John S. (Jay) III Gallagher, University of Wisconsin-Madison