Let me build up to that slowly.
An object's gravitational monopole is just the total amount of its mass.
An object's gravitational dipole is a measure of how much that mass is distributed away from some center in some direction. It's a vector, since it had to convey not only how much the mass is off-center but also which way. Considering some object in the abstract, the natural 'center' to pick is the center of mass, which is the point around which the dipole is zero.
The quadrupole represents how stretched-out along some axis the mass is. A sphere has zero quadrupole. A rod has a quadrupole. A flat disk also has a quadrupole, with the opposite sign of the quadrupole of a rod pointing out from its flat sides. The rod is a sphere stretched along that axis and the disk is a sphere squashed along that axis. In general, objects can have quadrupole moments along three different axes at right angles to each other. (The quadrupole moment is something called a tensor.)
The quadrupole moment can definitely change. Think of two balls attached by a spring. If they are stretched apart and then allowed to oscillate, the quadrupole moment will get smaller and bigger in the oscillations.
The quadrupole moment does give gravitational fields, but they fall off much faster as you leave the object than does the main monopole field, which falls as the square of the distance from the center. The quadrupole field falls as the fourth power of the distance.
(republished on 07/12/06)