A gravitational quadrupole is a mathematical concept to explain the gravitational effect of a mass which is not just a point but which has an extension in the space
The most easy way to show its effects is considering 4 equal masses (m each of them) placed in every corner of a square of side
By using the Newton's gravitational law
the same for the distance from mass M to every corner of square
the total gravitational force in the system is
Taken in account the approximation formula
In this expression, we have 2 elements. The first one
concentrated in a single point, just in the center of the square. The
quadropole for this example. See that this element is negative and decrease faster with distances.
This concept can be generalised to any geometrical configuration with 4 masses and also to larger number of masses (octopole, hexapole, etc.). When we have a extensive mass, its gravitational effects can be approximated in this way (point mass plus gravitational dipole plus gravitational quadrupole plus ...).