the addition of angular momentum

When two sources of angular momentum share the same system -- an electron's orbital motion and its spin, or the spins of two electrons in a bond -- you often need the total, J = J_1 + J_2, because it is the total that a spherically symmetric interaction conserves. But adding angular momenta quantum-mechanically is not like adding vectors classically: because the pieces cannot each point in a definite direction, the total magnitude itself is quantized, and the question 'what is the combined angular momentum?' has several allowed answers at once.

Combine an angular momentum j_1 with an angular momentum j_2. The total j can take every value from |j_1 - j_2| up to j_1 + j_2 in integer steps: j = |j_1 - j_2|, |j_1 - j_2| + 1, ..., j_1 + j_2. For each such j the projection m runs from -j to +j, and the accounting always balances: the total number of states, sum over j of (2j+1), equals (2 j_1 + 1)(2 j_2 + 1), the size of the original product space. Physically you are changing basis. The 'uncoupled' basis labels states by (m_1, m_2) -- each piece's projection separately; the 'coupled' basis labels them by (j, m) -- the total magnitude and projection. Both describe the same states; which is 'good' depends on the Hamiltonian.

The choice of basis is not academic. When two angular momenta are coupled by an interaction like spin-orbit coupling (proportional to L dot S), the coupled basis diagonalizes it, because L dot S = (J^2 - L^2 - S^2)/2 is diagonal in states of definite j -- this is exactly what produces the fine structure of atomic spectra. Adding two spin-1/2 electrons gives j = 0 (a single antisymmetric 'singlet') or j = 1 (a symmetric 'triplet' of three states), the distinction that underlies the chemical bond and the exchange interaction behind magnetism.