We know that J is the vector sum of L and S. L and S can take 2L+1 and 2S+1 orientations, respectively. Depending of the relative orientation of L and S, we obtain different orientations of J corresponding to the values J = L+S, L+S-1, …, |L-S|. In the situation of the picture L=1 and S=1, so that J can take three different orientations corresponding to the values J= 2, 1, 0.
If the vectors L and S are parallel and pointing in opposite directions, then J is the zero vector because L and S have the same magnitude. In the picture this corresponds to the bottom energy level with J = L-S = 0.
If L and S are parallel and pointing in the same direction, then J = L+S = 2.
If L and S are perpendicular to each other, then J = 1.