Let us say we have a Helium atom with two electrons. Total angular momentum of the system is a constant of motion. Assume that the electrons and protons do not have spin and there is only interaction between the protons and electrons. Assume two cases:
(1) The electrons do not interact. In this scenario, the angular momentum of each electron is a constant of motion.
(2) The electrons interact. In this scenario while the total angular momentum is a constant of motion as usual, the individual angular momentums are not. This is because if an electron increases its radius (thus increasing its angular momentum) the other electron has to reduce its radius so that total angular momentum remains constant.
In scenario (2) we say that the two angular momentums have coupled with each other. The interaction between the electrons introduces a “coupled” term in the Hamiltonian. So now the Hamiltonian is not commutative with the individual angular momentums any more.
Another good way to understand angular momentum coupling is to think of spin-orbit coupling in the Hydrogen Atom. This is a coupling between the intrinsic angular momentum (spin) of the electron and its orbital angular momentum. The mechanism of the coupling arises in the following fashion:
An electron in its rest frame sees the proton rotate around it. This rotating proton is a current that exerts a magnetic field on the electron. The electron has a dipole moment (by virtue of having spin) so it interacts with the magnetic field induced by the proton rotation. In the Hamiltonian we get a term involving the orbital and spin terms of the angular momentum. Now 
and 
are not longer constants of motion because of this “cross term”. Instead, we have 
as a constant of motion. (Why? ![\[H, \mathbf{J}\]=0 \Rightarrow](http://sidk.info/mimetex/pictures/03b245bc648739da2711bcdc9df566f8.gif "\[H, \mathbf{J}\]=0 \Rightarrow")
eigenstate of H is an eigenstate of 
. Lets say a system that does not exchange any energy is in some eigenstate. This is eigenstate corresponds to some specific . Thus is constant of motion).
Now because of the reasoning above, and are no longer constants of motion while is. Note that interestingly 
and 
are still constants of motion. (Note: 
). This makes sense because is a scalar. The magnitude of the intrinsic angular momentum of an electron never changes.
In conclusion, coupling is when two angular momentums via some interaction do not remain constants of motion anymore. Their values “couple” as they lose constancy but their sum remains constant as in above example.
Useful References:
(1) Griffiths, Introduction to Quantum Mechanics, 2nd Edition, pg 283
(2) http://en.wikipedia.org/wiki/Spin-orbit_coupling
(3) http://en.wikipedia.org/wiki/Angular_momentum_coupling
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