Features:

Gravity falls off with distance from the gravitational field source as an inverse-square-law (exactly if the source is spherically symmetric, approximately if NOT) obeying Newton's law of universal gravitation.
Thus, the gravitational force pulls unequally on the astro-body. The nearer parts of the body to the gravitational field source are pulled more strongly than the farther parts. The top panel of the image illustrates this.
If you subtract off the net gravitational force, you have the residual gravitational force trying to stretch the astro-body relative to its center of mass which moves under the net gravitational force only. Note the center of mass motion is NOT effected by INTERNAL forces: they cancel pairwise by Newton's 3rd law of motion in the classical limit (which we are assuming throughout the discussion in this figure).
The residual gravitational force is called the tidal force.
The lower panel of the image illustrates the tidal force and its stretching effect.
Of course, the tidal force just adds to INTERNAL forces acting on the astro-body (e.g., self-gravity, the pressure force, and the centrifugal force) and can be canceled by them usually after a distortion of the astro-body.
If the tidal force gets too strong relative to the INTERNAL forces, the astro-body can be disrupted. This happens to moons that get too close to their host planet.
Also the tidal force can prevent a planetary ring from coalecsing into a moon under its self-gravity.
In general, the tidal force:
1. Increases with the mass of the gravitational field source.
2. Decreases/Increases with increasing/decreasing distance to the gravitational field source.
3. Increases with the size of object acted on by the tidal force.
For a more elaborate explication of the tidal force, see Extended file: Mechanics file: tidal_force_4.html (which could be this file).