I'm currently reading about orbits of near-Earth satellites and some terminology is getting thrown around that I'm not sure I understand what they actually mean:
The Earth's monopole moment and the Earth's quadrupole moment?
What are some easily understood explanations of the above terms?
Answer
A monopole (gravitational) of a system is basically the amount of mass-energy the system has.
A dipole is a measure of how the mass is distributed away from some center.
The quadrupole moment describes how stretched out the mass distribution is along an axis. Quadrupole would be zero for a sphere, but non-zero for a rod, for instance. It is also non-zero for the Earth, because the Earth is an oblate spheroid.
The gravitational contribution from a quadrupole falls of faster than that of a monopole. (which is why the Earth's quadrupole moment is important for studying satellites and not really for studying the moon, owing to the $r^{-3}$ dependency of the contribution to the potential)
Quadrupoles and other higher order moments are important in GR because the change in their distribution can produce gravitational waves.
Example:
Let's consider two cases, in both the cases, the large bodies are of mass $M$ and the small one of mass $m$, and the small one is on the line of symmetry at a distance $r$.
Case 1: No quadrupole moment. 
The force here is a simple: $$\frac{GMm}{r^2}$$.
Case 2: Non-zero quadrupole moment. (the larger spheres are separated by some distance $2R$.)
The force in this case is: $$\frac{2GMmr}{(r^2+R^2)^{3/2}}$$
This, for large $r$, can be approximated to (two term series expansion): $$F \sim \frac{2GMm}{r^2}-\frac{3GMmR^2}{r^4}$$
The weird term here is because of the quadrupole moment of the system. As you go further away ($r>>R$), the force, $F$ is more or less: $$F \sim \frac{2GMm}{r^2}$$
This is why the "quadrupole moment effect" falls off with distance.
Apologies for the obnoxious MS Paint diagrams.
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