The total hamiltonian of a many electron system is a very complicated thing and we need to make a number of approximations to make the whole problem tractable. Anyway, here is what H would look like
H = sum((-h2/4pi2 m)deli2 + AiLi.Si + Ze2/ri) + (1/2)*sum (e2/rij) .
The first term inside the first term is the total kinetic energy, the second term is a spin-orbit interaction the third term is the potential energy due to nuclear interaction and the next sum denotes the potential energy due to electron-electron interaction. (The 1/2 is put in because when we sum over all i and all j we end up counting each ij pair twice) Of course its pretty hopeless to try and deal with this! So various things need to be done.
LS coupling is one of the approximation schemes that works for light atoms. Here we assume that the spin-orbit interaction is small compared with the total electron-electron interaction. It is clear that under this scheme, the total orbital angular momentum(L) will be conserved and so will the total spin angular momentum(S). This is what is meant by LS coupling. All the electronic orbital angular momenta couple to form a total L and all the spin angular momenta couple to form a total S which are individually conserved. Thus energy eigenstates of the Hamiltonian may be labelled by total L and total S.
This is the 0 order of perturbation theory and enough to explain the gross features of most spectra. Inner shells contribute neither to L nor to S and thus to determine the stationary states of the atom we just need to look at the valence electrons. This greatly simplifies matters.
The selection rules for transitions here become delta(L) = +1,-1 or 0 and not (0 to 0) and delta(S) = 0. The existence of energy levels and transitions which obey these selection rules provides evidence for LS coupling.
To explain the finer features of spectra we need to introduce spin-orbit interaction as a perturbation. Now neither L nor S is conserved but the total angular momentum J = L+S is conserved. Please look at this node on addition of angular momenta. Thus each LS state that we built up above gets split into a number of multiplets depending on the value of J. These multiplets are closely spaced and so only contribute to the fine splitting of spectral lines. Please look at this link for the notation used to describe these multiplets. The energy level of a multiplet(from the mean level of the LS level) may be calculated via
E = (const * Zeff/n3 ) * (j(j+1)-l(l+1)-s(s+1))/2 .
Thus the theory predicts that energy levels should be more closely spaced as n, the principal quantum number icreases and this prediction is borned out by experiment.
The other end of the scale where the spin orbit coupling term dominates compared to inter-electronic repulsion is called jj coupling.